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Probability is a powerful framework for quantifying uncertainty and randomness. In particular, conditional probability represents a way to update the uncertainty associated with an event given that specific information has been observed. For example, the probability that a person has a particular disease can be adjuste...	{ "Header 1": "2.4 Notes", "token_count": 561, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
[2.11](#page-438-0) Global warming. A Pew Research poll asked 1,306 Americans "From what you've read and heard, is there solid evidence that the average temperature on earth has been getting warmer over the past few decades, or not?". The table below shows the distribution of responses by party and ideology, where ...	{ "Header 1": "2.5 Exercises", "Header 3": "2.5.2 Conditional probability", "token_count": 1006, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
[2.13](#page-438-2) Seat belts. Seat belt use is the most effective way to save lives and reduce injuries in motor vehicle crashes. In a 2014 survey, respondents were asked, "How often do you use seat belts when you drive or ride in a car?". The following table shows the distribution of seat belt usage by sex. \| ...	{ "Header 1": "2.5 Exercises", "Header 3": "2.5. EXERCISES 131", "token_count": 1957, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
2.19 Views on evolution. A 2013 analysis conducted by the Pew Research Center found that 60% of survey respondents agree with the statement "humans and other living things have evolved over time" while 33% say that "humans and other living things have existed in their present form since the beginning of time" (7% r...	{ "Header 1": "2.5 Exercises", "Header 3": "2.5. EXERCISES", "token_count": 1639, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
2.26 Eye color. One of the earliest models for the genetics of eye color was developed in 1907, and proposed a single-gene inheritance model, for which brown eye color is always dominant over blue eye color. Suppose that in the population, 25% of individuals are homozygous dominant (BB), 50% are heterozygous (*Bb...	{ "Header 1": "2.5 Exercises", "Header 3": "2.5.3 Extended example", "token_count": 854, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
[2.29](#page-439-4) Genetics of Australian cattle dogs. Australian cattle dogs are known to have a high prevalence of congenital deafness. Deafness in both ears is referred to as bilateral deafness, while deafness in one ear is referred to as unilateral deafness. Deafness in dogs is associated with the white spot...	{ "Header 1": "2.5 Exercises", "Header 3": "2.5. EXERCISES 137", "token_count": 815, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
- [3.1](#page-139-0) [Random variables](#page-139-0) - [3.2](#page-146-0) [Binomial distribution](#page-146-0) - [3.3](#page-151-0) [Normal distribution](#page-151-0) - [3.4](#page-167-0) [Poisson distribution](#page-167-0) - *[3.5](#page-169-0) [Distributions related to Bernoulli trials](#page-169-0)...	{ "Header 1": "Distributions of random variables", "token_count": 627, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Formally, a random variable assigns numerical values to the outcome of a random phenomenon, and is usually written with a capital letter such as X, Y , or Z. If a coin is tossed three times, the outcome is the sequence of observed heads and tails. One such outcome might be TTH: tails on the first two tosses, he...	{ "Header 1": "3.1 Random variables", "Header 3": "3.1.1 Distributions of random variables", "token_count": 1369, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Just like distributions of data, distributions of random variables also have means, variances, standard deviations, medians, etc.; these characteristics are computed a bit differently for random variables. The mean of a random variable is called its expected value and written E(X). To calculate the mean of a random...	{ "Header 1": "3.1 Random variables", "Header 3": "3.1.2 Expectation", "token_count": 221, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Calculate the expected value of X, where X represents the number of heads in three tosses of a fair coin. X can take on values 0, 1, 2, and 3. The probability of each $x_k$ is given in Figure 3.2. E E(X) Expected Value of X $$E(X) = x_1 P(X = x_1) + \dots + x_k P(X = x_k)$$ = (0)(P(X = 0)) + (1)(P(X =...	{ "Header 1": "3.1 Random variables", "Header 3": "EXAMPLE 3.2", "token_count": 281, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Calculate the expected value of Y, where Y represents the number of heads in three tosses of an unfair coin, where the probability of heads is 0.70.<sup>4</sup> <sup>&</sup>lt;sup>3</sup>The expected value E(X) can also be expressed as $\mu$ , e.g. $\mu = 1.5$ <sup>&</sup>lt;sup>4</sup>First, calculate the pr...	{ "Header 1": "3.1 Random variables", "Header 3": "GUIDED PRACTICE 3.3", "token_count": 323, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
The variability of a random variable can be described with variance and standard deviation. For data, the variance is computed by squaring deviations from the mean (xi−µ) and then averaging over the number of values in the dataset (Section [1.4.2\)](#page-30-0). In the case of a random variable, the squared devia...	{ "Header 1": "3.1 Random variables", "Header 3": "3.1.3 Variability of random variables", "token_count": 434, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Compute the variance and standard deviation of X, the number of heads in three tosses of a fair coin. In the formula for the variance, k = 4 and µ<sup>X</sup> = E(X) = 1.5. $$\sigma_X^2 = (x_1 - \mu_X)^2 P(X = x_1) + \dots + (x_4 - \mu)^2 P(X = x_4)$$ = (0 - 1.5)<sup>2</sup>(1/8) + (1 - 1.5)<sup>2</su...	{ "Header 1": "3.1 Random variables", "Header 3": "EXAMPLE 3.5", "token_count": 575, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Ignoring the cost of prescription drugs, over-the-counter medications, and the annual deductible amount, calculate the expectation and the standard deviation of the expected annual health care cost for this employee. Let the random variable X denote annual health care costs, where $x_i$ represents the costs in a ye...	{ "Header 1": "3.1 Random variables", "Header 3": "EXAMPLE 3.6", "token_count": 901, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
If X and Y are random variables, then a linear combination of the random variables is given by aX + bY , where a and b are constants. The mean of a linear combination of random variables is $$E(aX + bY) = aE(X) + bE(Y) = a\mu_X + b\mu_Y.$$ The formula easily generalizes to a sum of any number of ran...	{ "Header 1": "3.1 Random variables", "Header 3": "LINEAR COMBINATIONS OF RANDOM VARIABLES AND THEIR EXPECTED VALUES", "token_count": 413, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Suppose the employee will begin a domestic partnership in the next year. Although she and her companion will begin living together and sharing expenses, they will each keep their existing health insurance plans; both, in fact, have the same plan from the same employer. In the last five years, her partner visited a phys...	{ "Header 1": "3.1 Random variables", "Header 3": "GUIDED PRACTICE 3.7", "token_count": 472, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Calculate the variance and standard deviation for the combined cost of next year's health care for the two partners, assuming that the costs for each person are independent. Let X represent the sum of costs for the employee and Y the sum of costs for her partner. First, calculate the variance of health care cos...	{ "Header 1": "3.1 Random variables", "Header 3": "EXAMPLE 3.8", "token_count": 470, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Psychologist Stanley Milgram began a series of experiments in 1963 to study the effect of authority on obedience. In a typical experiment, a participant would be ordered by an authority figure to give a series of increasingly severe shocks to a stranger. Milgram found that only about 35% of people would resist the auth...	{ "Header 1": "3.2 Binomial distribution", "Header 3": "3.2.1 Bernoulli distribution", "token_count": 641, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
If X is a random variable that takes value 1 with probability of success p and 0 with probability 1 − p, then X is a Bernoulli random variable with mean p and standard deviation p p(1 − p). Suppose X represents the outcome of a single toss of a fair coin, where heads is labeled success. X is a Berno...	{ "Header 1": "3.2 Binomial distribution", "Header 3": "BERNOULLI RANDOM VARIABLE", "token_count": 256, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Suppose that four individuals are randomly selected to participate in Milgram's experiment. What is the chance that there will be exactly one successful trial, assuming independence between trials? Suppose that the probability of success remains 0.35. Consider a scenario in which there is one success (i.e., one perso...	{ "Header 1": "3.2 Binomial distribution", "Header 3": "EXAMPLE 3.9", "token_count": 364, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
The Bernoulli distribution is unrealistic in all but the simplest of settings. However, it is a useful building block for other distributions. The binomial distribution describes the probability of having exactly k successes in n independent Bernoulli trials with probability of a success p. In Example 3.9, the goal...	{ "Header 1": "3.2 Binomial distribution", "Header 3": "3.2.2 The binomial distribution", "token_count": 711, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Suppose the probability of a single trial being a success is p. The probability of observing exactly k successes in n independent trials is given by Bin(n, p) Binomial dist. with n trials & p prob. of success $$P(X=k) = \binom{n}{k} p^k (1-p)^{n-k} = \frac{n!}{k!(n-k)!} p^k (1-p)^{n-k}.$$ (3.11) Addit...	{ "Header 1": "3.2 Binomial distribution", "Header 3": "BINOMIAL DISTRIBUTION", "token_count": 206, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
What is the probability that 3 of 8 randomly selected participants will refuse to administer the worst shock? First, check the conditions for applying the binomial model. The number of trials is fixed (n = 8) and each trial outcome can be classified as either success or failure. The sample is random, so the trials ...	{ "Header 1": "3.2 Binomial distribution", "Header 3": "EXAMPLE 3.13", "token_count": 244, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
What is the probability that at most 3 of 8 randomly selected participants will refuse to administer the worst shock? The event of at most 3 out of 8 successes can be thought of as the combined probability of 0, 1, 2, and 3 successes. Thus, the probability that at most 3 of 8 will refuse is given by: (E) (E) (G...	{ "Header 1": "3.2 Binomial distribution", "Header 3": "EXAMPLE 3.14", "token_count": 356, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
The probability that a smoker will develop a severe lung condition in their lifetime is about 0.30. Suppose that 5 smokers are randomly selected from the population. What is the probability that (a) one will develop a severe lung condition? (b) that no more than one will develop a severe lung condition? (c) that at lea...	{ "Header 1": "3.2 Binomial distribution", "Header 3": "GUIDED PRACTICE 3.16", "token_count": 280, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
The normal distribution model always describes a symmetric, unimodal, bell-shaped curve. However, the curves can differ in center and spread; the model can be adjusted using mean and standard deviation. Changing the mean shifts the bell curve to the left or the right, while changing the standard deviation stretches or ...	{ "Header 1": "3.3 Normal distribution", "Header 3": "3.3.1 Normal distribution model", "token_count": 348, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
The Z-score of an observation quantifies how far the observation is from the mean, in units of Z standard deviation(s). If x is an observation from a distribution N(µ, σ), the Z-score is mathematically defined as: $$Z = \frac{x - \mu}{\sigma}.$$ An observation equal to the mean has a Z-score of 0. Observati...	{ "Header 1": "3.3 Normal distribution", "Header 3": "3.3.2 Standardizing with Z-scores", "token_count": 271, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
The SAT and the ACT are two standardized tests commonly used for college admissions in the United States. The distribution of test scores are both nearly normal. For the SAT, N(1500,300); for the ACT, N(21,5). While some colleges request that students submit scores from both tests, others allow students the cho...	{ "Header 1": "3.3 Normal distribution", "Header 3": "EXAMPLE 3.17", "token_count": 367, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Systolic blood pressure (SBP) for adults in the United States aged 18-39 follow an approximate normal distribution, N(115, 17.5). As age increases, systolic blood pressure also tends to increase. Mean systolic blood pressure for adults 60 years of age and older is 136 mm Hg, with standard deviation 40 mm Hg. Systolic b...	{ "Header 1": "3.3 Normal distribution", "Header 3": "GUIDED PRACTICE 3.19", "token_count": 458, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
The normal distribution is a continuous probability distribution. Recall from Section [2.1.5](#page-96-4) that the total area under the density curve is always equal to 1, and the probability that a variable has a value within a specified interval is the area under the curve over that interval. By using either statisti...	{ "Header 1": "3.3 Normal distribution", "Header 3": "3.3.4 Calculating normal probabilities", "token_count": 1716, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Cumulative SAT scores are well-approximated by a normal model, N(1500,300). What is the probability that a randomly selected test taker scores at least 1630 on the SAT? For any normal probability problem, it can be helpful to start out by drawing the normal curve and shading the area of interest. ![](_page_156_...	{ "Header 1": "3.3 Normal distribution", "Header 3": "EXAMPLE 3.22", "token_count": 348, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Recall that the probability of a continuous random variable equaling some exact value is always 0. As a result, for a continuous random variable X, P (X ≤ x) = P (X < x) and P (X ≥ x) = P (X > x). It is valid to state that P (X ≥ x) = 1 − P (X ≤ x) = 1 − P (X < x). This is not ...	{ "Header 1": "3.3 Normal distribution", "Header 3": "DISCRETE VERSUS CONTINUOUS PROBABILITIES", "token_count": 208, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Systolic blood pressure for adults 60 years of age and older in the United States is approximately normally distributed: N(136,40). What is the probability of an adult in this age group having systolic blood pressure of 140 mm Hg or greater?[15](#page-157-1) <sup>14</sup>This probability was calculated as part of...	{ "Header 1": "3.3 Normal distribution", "Header 3": "GUIDED PRACTICE 3.24", "token_count": 226, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
The height of adult males in the United States between the ages of 20 and 62 is nearly normal, with mean 70 inches and standard deviation 3.3 inches.[16](#page-158-0) What is the probability that a random adult male is between 5'9" and 6'2"? These heights correspond to 69 inches and 74 inches. First, draw the figure....	{ "Header 1": "3.3 Normal distribution", "Header 3": "EXAMPLE 3.25", "token_count": 348, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
What percentage of adults in the United States ages 60 and older have blood pressure between 145 and 130 mm Hg?[17](#page-158-1) Z<sup>130</sup> = −0.15 → 0.4404 (area above). Final answer: 0.5890 − 0.4404 = 0.1486. <sup>16</sup>As based on a sample of 100 men, from the USDA Food Commodity Intake Databa...	{ "Header 1": "3.3 Normal distribution", "Header 3": "GUIDED PRACTICE 3.26", "token_count": 203, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
How tall is a man with height in the 40th percentile? First, draw a picture. The lower tail probability is 0.40, so the shaded area must start before the mean. ![](_page_159_Figure_4.jpeg) Determine the Z-score associated with the 40th percentile. Because the percentile is below 50%, Z will be negative. Loo...	{ "Header 1": "3.3 Normal distribution", "Header 3": "EXAMPLE 3.27", "token_count": 266, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
(a) What is the 95th percentile for SAT scores? (b) What is the 97.5 th percentile of the male heights?[18](#page-159-0) <sup>18</sup>(a) Look for 0.95 in the probability portion (middle part) of the normal probability table: row 1.6 and (about) column 0.05, i.e. Z95 = 1.65. Knowing Z95 = 1.65, µ = 15...	{ "Header 1": "3.3 Normal distribution", "Header 3": "GUIDED PRACTICE 3.28", "token_count": 222, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
The normal distribution can be used to approximate other distributions, such as the binomial distribution. The binomial formula is cumbersome when sample size is large, particularly when calculating probabilities for a large number of observations. Under certain conditions, the normal distribution can be used to approx...	{ "Header 1": "3.3 Normal distribution", "Header 3": "3.3.6 Normal approximation to the binomial distribution", "token_count": 315, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Approximately 20% of the US population smokes cigarettes. A local government commissioned a survey of 400 randomly selected individuals to investigate whether their community might have a lower smoker rate than 20%. The survey found that 59 of the 400 participants smoke cigarettes. If the true proportion of smokers in ...	{ "Header 1": "3.3 Normal distribution", "Header 3": "EXAMPLE 3.29", "token_count": 797, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
The normal model can also be used to approximate data distributions. While using a normal model can be convenient, it is important to remember that normality is always an approximation. Testing the appropriateness of the normal assumption is a key step in many data analyses. ![](_page_162_Figure_3.jpeg) Figure 3.14...	{ "Header 1": "3.3 Normal distribution", "Header 3": "3.3.7 Evaluating the normal approximation", "token_count": 308, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Three datasets were simulated from a normal distribution, with sample sizes n = 40, n = 100, and n = 400; the histograms and normal probability plots of the datasets are shown in Figure 3.15. What happens as sample size increases? As sample size increases, the data more closely follows the normal distribution; the hi...	{ "Header 1": "3.3 Normal distribution", "Header 3": "EXAMPLE 3.30", "token_count": 275, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Determine which data sets represented in Figure 3.18 plausibly come from a nearly normal distribution.<sup>21</sup> ![](_page_165_Figure_3.jpeg) Figure 3.18: Four normal probability plots for Guided Practice 3.33. (G) <sup>&</sup>lt;sup>21</sup>Answers may vary. The top-left plot shows some deviations in the sm...	{ "Header 1": "3.3 Normal distribution", "Header 3": "GUIDED PRACTICE 3.33", "token_count": 267, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
The Poisson distribution is a probability model for the number of events that occur in a population. The probability that exactly k events occur is given by $$P(X=k) = \frac{e^{-\lambda}(\lambda)^k}{k!},$$ where k may take a value 0, 1, 2, ... The mean and standard deviation of this distribution are $\lambda$ ...	{ "Header 1": "3.4 Poisson distribution", "Header 3": "POISSON DISTRIBUTION", "token_count": 379, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
For children ages 0 - 14, the incidence rate of acute lymphocytic leukemia (ALL) was approximately 30 diagnosed cases per million children per year in 2010. Approximately 20% of the US population of 319,055,000 are in this age range. What is the expected number of cases of ALL in the US over five years? The incidence...	{ "Header 1": "3.4 Poisson distribution", "Header 3": "EXAMPLE 3.37", "token_count": 268, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
(E) Recall that in the Milgram shock experiments, the probability of a person refusing to give the most severe shock is p = 0.35. Suppose that participants are tested one at a time until one person refuses; i.e., until the first occurrence of a successful trial. What are the chances that the first occurrence happens ...	{ "Header 1": "3.5 Distributions related to Bernoulli trials", "Header 3": "EXAMPLE 3.38", "token_count": 321, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
What is the probability of the first success occurring within the first 4 people? This is the probability it is the first (k = 1), second (k = 2), third (k = 3), or fourth (k = 4) trial that is the first success, which represent four disjoint outcomes. Compute the probability of each case and add the separate...	{ "Header 1": "3.5 Distributions related to Bernoulli trials", "Header 3": "EXAMPLE 3.40", "token_count": 404, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Each sequence in Figure [3.22](#page-171-0) has exactly two failures and four successes with the last attempt always being a success. If the probability of a success is p = 0.8, find the probability of the first sequence.[26](#page-171-1) If the probability of a successful extraction is p = 0.8, what is the p...	{ "Header 1": "3.5 Distributions related to Bernoulli trials", "Header 3": "GUIDED PRACTICE 3.41", "token_count": 489, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
On 70% of days, a hospital admits at least one heart attack patient. On 30% of the days, no heart attack patients are admitted. Identify each case below as a binomial or negative binomial case, and compute the probability. (a) What is the probability the hospital will admit a heart attack patient on exactly three days ...	{ "Header 1": "3.5 Distributions related to Bernoulli trials", "Header 3": "GUIDED PRACTICE 3.45", "token_count": 561, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Suppose that a large number of deer live in a forest. Researchers are interested in using the capture-recapture method to estimate total population size. A number of deer are captured in an initial sample and marked, then released; at a later time, another sample of deer are captured, and the number of marked and unmar...	{ "Header 1": "3.5 Distributions related to Bernoulli trials", "Header 3": "3.5.3 Hypergeometric distribution", "token_count": 891, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
A small clinic would like to draw a random sample of 10 individuals from their patient list of 120, of which 30 patients are smokers. (a) What is the probability of 6 individuals in the sample being smokers? (b) What is the probability that at least 2 individuals in the sample smoke?<sup>31</sup> <sup>31</sup>(a) Let...	{ "Header 1": "3.5 Distributions related to Bernoulli trials", "Header 3": "GUIDED PRACTICE 3.47", "token_count": 367, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Example [3.8](#page-145-0) calculated the variability in health care costs for an employee and her partner relying on the assumption that the number of health episodes between the two are not related. It could be reasonable to assume that the health status of one person gives no information about the other's health, gi...	{ "Header 1": "3.6 Distributions for pairs of random variables", "token_count": 491, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
The joint distribution pX,Y (x,y) for a pair of random variables (X,Y ) is the collection of probabilities $$p(x_i, y_j) = P(X = x_i \text{ and } Y = y_j)$$ for all pairs of values (x<sup>i</sup> ,yj ) that the random variables X and Y take on. Joint distributions are often displayed in tabular form as ...	{ "Header 1": "3.6 Distributions for pairs of random variables", "Header 3": "JOINT DISTRIBUTION", "token_count": 1140, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
(E) If it is known that the employee's annual health care cost is \$968, what is the conditional distribution of the partner's annual health care cost? Note that there is a different conditional distribution of Y for every possible value of X; this problem specifically asks for the conditional distribution of *...	{ "Header 1": "3.6 Distributions for pairs of random variables", "Header 3": "EXAMPLE 3.48", "token_count": 307, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Consider two random variables, X and Y, with the joint distribution shown in Figure 3.27. - (a) Compute the marginal distributions of X and Y. - (b) Identify the joint probability $p_{X,Y}(1,2)$ . - (c) What is the value of $p_{X,Y}(2,1)$ ? - (d) Compute the conditional distribution of X given that Y = 2....	{ "Header 1": "3.6 Distributions for pairs of random variables", "Header 3": "GUIDED PRACTICE 3.49", "token_count": 806, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
(E) Demonstrate that the employee's health care costs and the partner's health care costs are not independent random variables. As shown in Example 3.48, the conditional distribution of the partner's annual health care cost given that the employee's annual cost is \$968 is P(Y = \$968\|X = \$968) = 0.60, P(Y = \$988...	{ "Header 1": "3.6 Distributions for pairs of random variables", "Header 3": "EXAMPLE 3.50", "token_count": 376, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Based on Figure 3.27, check whether X and Y are independent.<sup>34</sup> Two random variables that are not independent are called correlated random variables. The correlation between two random variables is a measure of the strength of the relationship between them, just as it was for pairs of data points explor...	{ "Header 1": "3.6 Distributions for pairs of random variables", "Header 3": "GUIDED PRACTICE 3.51", "token_count": 859, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Compute the correlation between annual health care costs for the employee and her partner. As calculated previously, E(X) = \$1010, Var(X) = 1556, E(Y) = \$980, and Var(Y) = 96. Thus, SD(X) = \$39.45 and SD(Y) = \$9.80. $$\begin{split} \rho_{X,Y} &= p(x_1,y_1) \frac{(x_1 - \mu_X)}{\mathrm{sd}(X)} \frac{(y_1 - \mu_Y...	{ "Header 1": "3.6 Distributions for pairs of random variables", "Header 3": "EXAMPLE 3.55", "token_count": 547, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Based on Figure 3.27, compute the correlation between X and Y. For your convenience, the following values are provided: E(X) = 2.2, Var(X) = 2.16, E(Y) = 1.5, Var(Y) = 0.25.<sup>35</sup> When two random variables X and Y are correlated: Variance( $$X + Y$$ ) = Variance( $X$ ) + Variance( $Y$ ) + $2\sigma_X...	{ "Header 1": "3.6 Distributions for pairs of random variables", "Header 3": "GUIDED PRACTICE 3.56", "token_count": 264, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
The Association of American Medical Colleges (AAMC) introduced a new version of the Medical College Admission Test (MCAT) in the spring of 2015. Data from the scores were recently released by AAMC.[37](#page-182-0) The test consists of 4 components: chemical and physical foundations of biological systems; critical anal...	{ "Header 1": "3.6 Distributions for pairs of random variables", "Header 3": "EXAMPLE 3.61", "token_count": 529, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Thinking in terms of random variables and distributions of probabilities makes it easier to describe all possible outcomes of an experiment or process of interest, versus only considering probabilities on the scale of individual outcomes or sets of outcomes. Several of the fundamental concepts of probability can natura...	{ "Header 1": "3.7 Notes", "token_count": 313, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
[3.7](#page-442-0) Underage drinking, Part I. Data collected by the Substance Abuse and Mental Health Services Administration (SAMSHA) suggests that 69.7% of 18-20 year olds consumed alcoholic beverages in any given year.[38](#page-185-0) - (a) Suppose a random sample of ten 18-20 year olds is taken. Is the use o...	{ "Header 1": "3.8 Exercises", "Header 3": "3.8.2 Binomial distribution", "token_count": 2043, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
[3.37](#page-443-8) Computing Poisson probabilities. This is a simple exercise in computing probabilities for a Poisson random variable. Suppose that X is a Poisson random variable with rate parameter λ = 2. Calculate P (X = 2), P (X ≤ 2), and P (X ≥ 3). **[3.38](#page-443-9) Stenographer's typos....	{ "Header 1": "3.8 Exercises", "Header 3": "3.8.4 Poisson distribution", "token_count": 1077, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
[3.50](#page-444-1) Defective rate. A machine that produces a special type of transistor (a component of computers) has a 2% defective rate. The production is considered a random process where each transistor is independent of the others. - (a) What is the probability that the 10th transistor produced is the fi...	{ "Header 1": "3.8 Exercises", "Header 3": "3.8. EXERCISES 195", "token_count": 1277, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
- [4.1](#page-200-0) [Variability in estimates](#page-200-0) - [4.2](#page-204-0) [Confidence intervals](#page-204-0) - [4.3](#page-211-0) [Hypothesis testing](#page-211-0) - [4.4](#page-224-0) [Notes](#page-224-0) - [4.5](#page-226-0) [Exercises](#page-226-0) Not surprisingly, many studies are no...	{ "Header 1": "Foundations for inference", "token_count": 1637, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
How would one estimate the difference in average weight between men and women? Given that xmen = 185.1 lbs and xwomen = 162.3 lbs, what is a good point estimate for the population difference?[7](#page-200-2) Point estimates become more accurate with increasing sample size. Figure [4.3](#page-200-3) shows the ...	{ "Header 1": "4.1 Variability in estimates", "Header 3": "GUIDED PRACTICE 4.1", "token_count": 480, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
The sample mean weight calculated from cdc.samp is 173.3 lbs. Another random sample of 60 participants might produce a different value of x, such as 169.5 lbs; repeated random sampling could result in additional different values, perhaps 172.1 lbs, 168.5 lbs, and so on. Each sample mean x can be thought of as a sin...	{ "Header 1": "4.1 Variability in estimates", "Header 3": "4.1.1 The sampling distribution for the mean", "token_count": 263, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
If a sample consists of at least 30 independent observations and the data are not strongly skewed, then the distribution of the sample mean is well approximated by a normal model. ![](_page_202_Figure_1.jpeg) Figure 4.5: The left panel shows a histogram of the sample means for 100,000 random samples. The right pane...	{ "Header 1": "4.1 Variability in estimates", "Header 3": "CENTRAL LIMIT THEOREM, INFORMAL DESCRIPTION", "token_count": 223, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
The standard error (SE) of the sample mean measures the sample-to-sample variability of X, error the extent to which values of the repeated sample means oscillate around the population mean. The theoretical standard error of the sample mean is calculated by dividing the population standard deviation (σ<sup>x</sup> ...	{ "Header 1": "4.1 Variability in estimates", "Header 3": "4.1.2 Standard error of the mean", "token_count": 482, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
- The population mean and standard deviation are denoted by µ and σ. - The sample mean and standard deviation are denoted by x and s. - The distribution of the random variable X refers to the collection of sample means if multiple samples of the same size were repeatedly drawn from a population. - The mean of...	{ "Header 1": "4.1 Variability in estimates", "Header 3": "SUMMARY: POINT ESTIMATE TERMINOLOGY", "token_count": 232, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
While a point estimate consists of a single value, an interval estimate provides a plausible range of values for a parameter. When estimating a population mean µ, a confidence interval for µ has the general form $$(\overline{x}-m, \overline{x}+m) = \overline{x} \pm m,$$ where m is the margin of error. Interva...	{ "Header 1": "4.2 Confidence intervals", "Header 3": "4.2.1 Interval estimates for a population parameter", "token_count": 766, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
The sample mean adult weight from the 60 observations in cdc.samp is xweight = 173.3 lbs, and the standard deviation is sweight = 49.04 lbs. Use Equation [4.2](#page-204-2) to calculate an approximate 95% confidence interval for the average adult weight in the US population. The standard error for the sample ...	{ "Header 1": "4.2 Confidence intervals", "Header 3": "EXAMPLE 4.3", "token_count": 345, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Let Y be a normally distributed random variable. Ninety-nine percent of the time, Y will be within how many standard deviations of the mean? This is equivalent to the z-score with 0.005 area to the right of z and 0.005 to the left of −z. In the normal probability table, this is the z-value that with 0.005...	{ "Header 1": "4.2 Confidence intervals", "Header 3": "EXAMPLE 4.6", "token_count": 585, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Create a 99% confidence interval for the average adult weight in the US population using the data in cdc.samp. The point estimate is xweight = 173.3 and the standard error is SE<sup>x</sup> = 6.33. Apply the 99% confidence interval formula: xweight ± 2.58×SE<sup>x</sup> → (156.97,189.63). A data a...	{ "Header 1": "4.2 Confidence intervals", "Header 3": "EXAMPLE 4.8", "token_count": 605, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
One-sided confidence intervals for a population mean provide either a lower bound or an upper bound, but not both. One-sided confidence intervals have the form $(\overline{x} - m, \infty)$ or $(-\infty, \overline{x} + m)$ . While the margin of error m for a one-sided interval is still calculated from the standa...	{ "Header 1": "4.2 Confidence intervals", "Header 3": "4.2.3 One-sided confidence intervals", "token_count": 268, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
The correct interpretation of an XX% confidence interval is, "We are XX% confident that the population parameter is between ..." While it may be tempting to say that a confidence interval captures the population parameter with a certain probability, this is a common error. The confidence level only quantifies how plaus...	{ "Header 1": "4.2 Confidence intervals", "Header 3": "4.2.4 Interpreting confidence intervals", "token_count": 457, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Body mass index (BMI) is one measure of body weight that adjusts for height. The National Health and Nutrition Examination Survey (NHANES) consists of a set of surveys and measurements conducted by the US CDC to assess the health and nutritional status of adults and children in the United States. The dataset nhanes.sam...	{ "Header 1": "4.2 Confidence intervals", "Header 3": "EXAMPLE 4.11", "token_count": 813, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Important decisions in science, such as whether a new treatment for a disease should be approved for the market, are primarily data-driven. For example, does a clinical study of a new cholesterol-lowering drug provide robust evidence of a beneficial effect in patients at risk for heart disease? A confidence interval ca...	{ "Header 1": "4.3 Hypothesis testing", "token_count": 239, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
The null hypothesis (H0) often represents either a skeptical perspective or a claim to be tested. The alternative hypothesis (HA) is an alternative claim and is often represented by a range of possible parameter values. Generally, an investigator suspects that the null hypothesis is not true and performs a hypoth...	{ "Header 1": "4.3 Hypothesis testing", "Header 3": "NULL AND ALTERNATIVE HYPOTHESES", "token_count": 218, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
The claim to be tested is that the population average of the difference between actual and desired weight for US adults is equal to 0. $$H_0: \mu = 0.$$ In the absence of prior evidence that people typically wish to be lighter (or heavier), it is reasonable to begin with an alternative hypothesis that allows for di...	{ "Header 1": "4.3 Hypothesis testing", "Header 3": "Step 1: Formulating null and alternative hypotheses", "token_count": 354, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Calculating the test statistic t is analogous to standardizing observations with Z-scores as discussed in Chapter 3. The test statistic quantifies the number of standard deviations between the sample mean x and the population mean µ: $$t = \frac{\overline{x} - \mu_0}{s/\sqrt{n}},$$ where s is the sample sta...	{ "Header 1": "4.3 Hypothesis testing", "Header 3": "Step 3: Calculating the test statistic", "token_count": 230, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
The p-value is the probability of observing a sample mean as or more extreme than the observed value, under the assumption that the null hypothesis is true. In samples of size 40 or more, the t-statistic will have a standard normal distribution unless the data are strongly skewed or extreme outliers are present. Re...	{ "Header 1": "4.3 Hypothesis testing", "Header 3": "Step 4: Calculating the p-value", "token_count": 556, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
To reach a conclusion about the null hypothesis, directly compare p and α. Note that for a conclusion to be informative, it must be presented in the context of the original question; it is not useful to only state whether or not H<sup>0</sup> is rejected. If p > α, the observed sample mean is not extreme enou...	{ "Header 1": "4.3 Hypothesis testing", "Header 3": "Step 5: Drawing a conclusion", "token_count": 270, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Suppose that the mean weight difference in the sampled group of 60 adults had been 7 pounds instead of 18.2 pounds, but with the same standard deviation of 33.46 pounds. Would there still be enough evidence at the α = 0.05 level to reject H<sup>0</sup> : µ = 0 in favor of H<sup>A</sup> : µ , 0?[15](#page-21...	{ "Header 1": "4.3 Hypothesis testing", "Header 3": "GUIDED PRACTICE 4.12", "token_count": 267, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
While fish and other types of seafood are important for a healthy diet, nearly all fish and shellfish contain traces of mercury. Dietary exposure to mercury can be particularly dangerous for young children and unborn babies. Regulatory organizations such as the US Food and Drug Administration (FDA) provide guidelines a...	{ "Header 1": "4.3 Hypothesis testing", "Header 3": "EXAMPLE 4.13", "token_count": 1060, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
In 2015, the National Sleep Foundation published new guidelines for the amount of sleep recommended for adults: 7-9 hours of sleep per night.[18](#page-216-1) The NHANES survey includes a question asking respondents about how many hours per night they sleep; the responses are available in nhanes.samp. In the sample of ...	{ "Header 1": "4.3 Hypothesis testing", "Header 3": "EXAMPLE 4.14", "token_count": 609, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
The relationship between a hypothesis test and the corresponding confidence interval is defined by the significance level α; the two approaches are based on the same inferential logic, and differ only in perspective. The hypothesis testing approach asks whether x is far enough away from µ<sup>0</sup> to be consid...	{ "Header 1": "4.3 Hypothesis testing", "Header 3": "4.3.3 Hypothesis testing and confidence intervals", "token_count": 666, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Calculate the confidence interval for the average mercury level for bluefin tuna caught off the coast of New Jersey. The summary statistics for the sample of 21 fish are $\bar{x} = 0.53$ ppm and s = 0.16 ppm. Does the interval agree with the results of Example 4.13? The 95% confidence interval is: $$\overline{x} ...	{ "Header 1": "4.3 Hypothesis testing", "Header 3": "EXAMPLE 4.16", "token_count": 593, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Previously, a hypothesis test was conducted at $\alpha = 0.05$ to test the null hypothesis $H_0: \mu = 7$ hours against the alternative $H_A: \mu < 7$ hours, for the average sleep per night US adults. Calculate the corresponding one-sided confidence interval and compare the information obtained from a confidence ...	{ "Header 1": "4.3 Hypothesis testing", "Header 3": "EXAMPLE 4.17", "token_count": 504, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Hypothesis tests can potentially result in incorrect decisions, such as rejecting the null hypothesis when the null is actually true. Figure 4.17 shows the four possible ways that the conclusion of a test can be right or wrong. \| \| \| Test conclusion \| ...	{ "Header 1": "4.3 Hypothesis testing", "Header 3": "4.3.4 Decision errors", "token_count": 276, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
In a trial, the defendant is either innocent (H0) or guilty (HA). After hearing evidence from both the prosecution and the defense, the court must reach a verdict. What does a Type I Error represent in this context? What does a Type II Error represent? If the court makes a Type I error, this means the defendant i...	{ "Header 1": "4.3 Hypothesis testing", "Header 3": "EXAMPLE 4.18", "token_count": 300, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Reducing the error probability of one type of error increases the chance of making the other type. As a result, the significance level is often adjusted based on the consequences of any decisions that might follow from the result of a significance test. By convention, most scientific studies use a significance level ...	{ "Header 1": "4.3 Hypothesis testing", "Header 3": "Choosing a significance level", "token_count": 560, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
In some cases, the choice of a one-sided or two-sided test can influence whether the null hypothesis is rejected. For example, consider a sample for which the t-statistic is 1.80. If a twosided test is conducted at α = 0.05, the p-value is $$P(Z \le -\|t\|) + P(Z \ge \|t\|) = 2P(Z \ge 1.80) = 0.072.$$ There is ...	{ "Header 1": "4.3 Hypothesis testing", "Header 3": "4.3.5 Choosing between one-sided and two-sided tests", "token_count": 1166, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Formal hypothesis tests are designed for settings where a decision or a claim about a hypothesis follows a test, such as in scientific publications where an investigator wishes to claim that an intervention changes an outcome. However, progress in science is usually based on a collection of studies or experiments, and ...	{ "Header 1": "4.3 Hypothesis testing", "Header 3": "4.3.6 The informal use of p-values", "token_count": 286, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Confidence intervals and hypothesis testing are two of the central concepts in inference for a population based on a sample. The confidence interval shows a range of population parameter values consistent with the observed sample, and is often used to design additional studies. Hypothesis testing is a useful tool for e...	{ "Header 1": "4.4 Notes", "token_count": 874, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
[4.4](#page-445-4) Mental health, Part I. The 2010 General Social Survey asked the question: "For how many days during the past 30 days was your mental health, which includes stress, depression, and problems with emotions, not good?" Based on responses from 1,151 US residents, the survey reported a 95% confidence i...	{ "Header 1": "4.5 Exercises", "Header 3": "4.5.2 Confidence intervals", "token_count": 1259, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
[4.8](#page-446-1) Age at first marriage, Part I. The National Survey of Family Growth conducted by the Centers for Disease Control gathers information on family life, marriage and divorce, pregnancy, infertility, use of contraception, and men's and women's health. One of the variables collected on this survey is t...	{ "Header 1": "4.5 Exercises", "Header 3": "4.5. EXERCISES 231", "token_count": 1184, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
[4.13](#page-446-6) Online communication. A study suggests that the average college student spends 10 hours per week communicating with others online. You believe that this is an underestimate and decide to collect your own sample for a hypothesis test. You randomly sample 60 students from your dorm and find that o...	{ "Header 1": "4.5 Exercises", "Header 3": "4.5. EXERCISES 233", "token_count": 1612, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
[4.21](#page-447-2) Testing for fibromyalgia. A patient named Diana was diagnosed with fibromyalgia, a long-term syndrome of body pain, and was prescribed anti-depressants. Being the skeptic that she is, Diana didn't initially believe that anti-depressants would help her symptoms. However after a couple months of b...	{ "Header 1": "4.5 Exercises", "Header 3": "4.5. EXERCISES 235", "token_count": 805, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
The tools studied in Chapter [4](#page-197-0) all made use of the t-statistic from a sample mean, $$t = \frac{\overline{x} - \mu}{s/\sqrt{n}},$$ where the parameter µ is a population mean, x and s are the sample mean and standard deviation, and n is the sample size. Tests and confidence intervals were res...	{ "Header 1": "5.1 Single-sample inference with the t-distribution", "token_count": 280, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Figure [5.1](#page-237-1) shows a t-distribution and normal distribution. Like the standard normal distribution, the t-distribution is unimodal and symmetric about zero. However, the tails of a t-distribution are thicker than for the normal, so observations are more likely to fall beyond two standard deviations f...	{ "Header 1": "5.1 Single-sample inference with the t-distribution", "Header 3": "5.1.1 The t-distribution", "token_count": 431, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }

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Probability is a powerful framework for quantifying uncertainty and randomness. In particular, conditional probability represents a way to update the uncertainty associated with an event given that specific information has been observed. For example, the probability that a person has a particular disease can be adjuste...	{ "Header 1": "2.4 Notes", "token_count": 561, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
[2.11](#page-438-0) Global warming. A Pew Research poll asked 1,306 Americans "From what you've read and heard, is there solid evidence that the average temperature on earth has been getting warmer over the past few decades, or not?". The table below shows the distribution of responses by party and ideology, where ...	{ "Header 1": "2.5 Exercises", "Header 3": "2.5.2 Conditional probability", "token_count": 1006, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
[2.13](#page-438-2) Seat belts. Seat belt use is the most effective way to save lives and reduce injuries in motor vehicle crashes. In a 2014 survey, respondents were asked, "How often do you use seat belts when you drive or ride in a car?". The following table shows the distribution of seat belt usage by sex. \| ...	{ "Header 1": "2.5 Exercises", "Header 3": "2.5. EXERCISES 131", "token_count": 1957, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
2.19 Views on evolution. A 2013 analysis conducted by the Pew Research Center found that 60% of survey respondents agree with the statement "humans and other living things have evolved over time" while 33% say that "humans and other living things have existed in their present form since the beginning of time" (7% r...	{ "Header 1": "2.5 Exercises", "Header 3": "2.5. EXERCISES", "token_count": 1639, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
2.26 Eye color. One of the earliest models for the genetics of eye color was developed in 1907, and proposed a single-gene inheritance model, for which brown eye color is always dominant over blue eye color. Suppose that in the population, 25% of individuals are homozygous dominant (BB), 50% are heterozygous (*Bb...	{ "Header 1": "2.5 Exercises", "Header 3": "2.5.3 Extended example", "token_count": 854, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
[2.29](#page-439-4) Genetics of Australian cattle dogs. Australian cattle dogs are known to have a high prevalence of congenital deafness. Deafness in both ears is referred to as bilateral deafness, while deafness in one ear is referred to as unilateral deafness. Deafness in dogs is associated with the white spot...	{ "Header 1": "2.5 Exercises", "Header 3": "2.5. EXERCISES 137", "token_count": 815, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
- [3.1](#page-139-0) [Random variables](#page-139-0) - [3.2](#page-146-0) [Binomial distribution](#page-146-0) - [3.3](#page-151-0) [Normal distribution](#page-151-0) - [3.4](#page-167-0) [Poisson distribution](#page-167-0) - *[3.5](#page-169-0) [Distributions related to Bernoulli trials](#page-169-0)...	{ "Header 1": "Distributions of random variables", "token_count": 627, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Formally, a random variable assigns numerical values to the outcome of a random phenomenon, and is usually written with a capital letter such as X, Y , or Z. If a coin is tossed three times, the outcome is the sequence of observed heads and tails. One such outcome might be TTH: tails on the first two tosses, he...	{ "Header 1": "3.1 Random variables", "Header 3": "3.1.1 Distributions of random variables", "token_count": 1369, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Just like distributions of data, distributions of random variables also have means, variances, standard deviations, medians, etc.; these characteristics are computed a bit differently for random variables. The mean of a random variable is called its expected value and written E(X). To calculate the mean of a random...	{ "Header 1": "3.1 Random variables", "Header 3": "3.1.2 Expectation", "token_count": 221, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Calculate the expected value of X, where X represents the number of heads in three tosses of a fair coin. X can take on values 0, 1, 2, and 3. The probability of each $x_k$ is given in Figure 3.2. E E(X) Expected Value of X $$E(X) = x_1 P(X = x_1) + \dots + x_k P(X = x_k)$$ = (0)(P(X = 0)) + (1)(P(X =...	{ "Header 1": "3.1 Random variables", "Header 3": "EXAMPLE 3.2", "token_count": 281, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Calculate the expected value of Y, where Y represents the number of heads in three tosses of an unfair coin, where the probability of heads is 0.70.<sup>4</sup> <sup>&</sup>lt;sup>3</sup>The expected value E(X) can also be expressed as $\mu$ , e.g. $\mu = 1.5$ <sup>&</sup>lt;sup>4</sup>First, calculate the pr...	{ "Header 1": "3.1 Random variables", "Header 3": "GUIDED PRACTICE 3.3", "token_count": 323, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
The variability of a random variable can be described with variance and standard deviation. For data, the variance is computed by squaring deviations from the mean (xi−µ) and then averaging over the number of values in the dataset (Section [1.4.2\)](#page-30-0). In the case of a random variable, the squared devia...	{ "Header 1": "3.1 Random variables", "Header 3": "3.1.3 Variability of random variables", "token_count": 434, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Compute the variance and standard deviation of X, the number of heads in three tosses of a fair coin. In the formula for the variance, k = 4 and µ<sup>X</sup> = E(X) = 1.5. $$\sigma_X^2 = (x_1 - \mu_X)^2 P(X = x_1) + \dots + (x_4 - \mu)^2 P(X = x_4)$$ = (0 - 1.5)<sup>2</sup>(1/8) + (1 - 1.5)<sup>2</su...	{ "Header 1": "3.1 Random variables", "Header 3": "EXAMPLE 3.5", "token_count": 575, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Ignoring the cost of prescription drugs, over-the-counter medications, and the annual deductible amount, calculate the expectation and the standard deviation of the expected annual health care cost for this employee. Let the random variable X denote annual health care costs, where $x_i$ represents the costs in a ye...	{ "Header 1": "3.1 Random variables", "Header 3": "EXAMPLE 3.6", "token_count": 901, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
If X and Y are random variables, then a linear combination of the random variables is given by aX + bY , where a and b are constants. The mean of a linear combination of random variables is $$E(aX + bY) = aE(X) + bE(Y) = a\mu_X + b\mu_Y.$$ The formula easily generalizes to a sum of any number of ran...	{ "Header 1": "3.1 Random variables", "Header 3": "LINEAR COMBINATIONS OF RANDOM VARIABLES AND THEIR EXPECTED VALUES", "token_count": 413, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Suppose the employee will begin a domestic partnership in the next year. Although she and her companion will begin living together and sharing expenses, they will each keep their existing health insurance plans; both, in fact, have the same plan from the same employer. In the last five years, her partner visited a phys...	{ "Header 1": "3.1 Random variables", "Header 3": "GUIDED PRACTICE 3.7", "token_count": 472, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Calculate the variance and standard deviation for the combined cost of next year's health care for the two partners, assuming that the costs for each person are independent. Let X represent the sum of costs for the employee and Y the sum of costs for her partner. First, calculate the variance of health care cos...	{ "Header 1": "3.1 Random variables", "Header 3": "EXAMPLE 3.8", "token_count": 470, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Psychologist Stanley Milgram began a series of experiments in 1963 to study the effect of authority on obedience. In a typical experiment, a participant would be ordered by an authority figure to give a series of increasingly severe shocks to a stranger. Milgram found that only about 35% of people would resist the auth...	{ "Header 1": "3.2 Binomial distribution", "Header 3": "3.2.1 Bernoulli distribution", "token_count": 641, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
If X is a random variable that takes value 1 with probability of success p and 0 with probability 1 − p, then X is a Bernoulli random variable with mean p and standard deviation p p(1 − p). Suppose X represents the outcome of a single toss of a fair coin, where heads is labeled success. X is a Berno...	{ "Header 1": "3.2 Binomial distribution", "Header 3": "BERNOULLI RANDOM VARIABLE", "token_count": 256, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Suppose that four individuals are randomly selected to participate in Milgram's experiment. What is the chance that there will be exactly one successful trial, assuming independence between trials? Suppose that the probability of success remains 0.35. Consider a scenario in which there is one success (i.e., one perso...	{ "Header 1": "3.2 Binomial distribution", "Header 3": "EXAMPLE 3.9", "token_count": 364, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
The Bernoulli distribution is unrealistic in all but the simplest of settings. However, it is a useful building block for other distributions. The binomial distribution describes the probability of having exactly k successes in n independent Bernoulli trials with probability of a success p. In Example 3.9, the goal...	{ "Header 1": "3.2 Binomial distribution", "Header 3": "3.2.2 The binomial distribution", "token_count": 711, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Suppose the probability of a single trial being a success is p. The probability of observing exactly k successes in n independent trials is given by Bin(n, p) Binomial dist. with n trials & p prob. of success $$P(X=k) = \binom{n}{k} p^k (1-p)^{n-k} = \frac{n!}{k!(n-k)!} p^k (1-p)^{n-k}.$$ (3.11) Addit...	{ "Header 1": "3.2 Binomial distribution", "Header 3": "BINOMIAL DISTRIBUTION", "token_count": 206, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
What is the probability that 3 of 8 randomly selected participants will refuse to administer the worst shock? First, check the conditions for applying the binomial model. The number of trials is fixed (n = 8) and each trial outcome can be classified as either success or failure. The sample is random, so the trials ...	{ "Header 1": "3.2 Binomial distribution", "Header 3": "EXAMPLE 3.13", "token_count": 244, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
What is the probability that at most 3 of 8 randomly selected participants will refuse to administer the worst shock? The event of at most 3 out of 8 successes can be thought of as the combined probability of 0, 1, 2, and 3 successes. Thus, the probability that at most 3 of 8 will refuse is given by: (E) (E) (G...	{ "Header 1": "3.2 Binomial distribution", "Header 3": "EXAMPLE 3.14", "token_count": 356, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
The probability that a smoker will develop a severe lung condition in their lifetime is about 0.30. Suppose that 5 smokers are randomly selected from the population. What is the probability that (a) one will develop a severe lung condition? (b) that no more than one will develop a severe lung condition? (c) that at lea...	{ "Header 1": "3.2 Binomial distribution", "Header 3": "GUIDED PRACTICE 3.16", "token_count": 280, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
The normal distribution model always describes a symmetric, unimodal, bell-shaped curve. However, the curves can differ in center and spread; the model can be adjusted using mean and standard deviation. Changing the mean shifts the bell curve to the left or the right, while changing the standard deviation stretches or ...	{ "Header 1": "3.3 Normal distribution", "Header 3": "3.3.1 Normal distribution model", "token_count": 348, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
The Z-score of an observation quantifies how far the observation is from the mean, in units of Z standard deviation(s). If x is an observation from a distribution N(µ, σ), the Z-score is mathematically defined as: $$Z = \frac{x - \mu}{\sigma}.$$ An observation equal to the mean has a Z-score of 0. Observati...	{ "Header 1": "3.3 Normal distribution", "Header 3": "3.3.2 Standardizing with Z-scores", "token_count": 271, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
The SAT and the ACT are two standardized tests commonly used for college admissions in the United States. The distribution of test scores are both nearly normal. For the SAT, N(1500,300); for the ACT, N(21,5). While some colleges request that students submit scores from both tests, others allow students the cho...	{ "Header 1": "3.3 Normal distribution", "Header 3": "EXAMPLE 3.17", "token_count": 367, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Systolic blood pressure (SBP) for adults in the United States aged 18-39 follow an approximate normal distribution, N(115, 17.5). As age increases, systolic blood pressure also tends to increase. Mean systolic blood pressure for adults 60 years of age and older is 136 mm Hg, with standard deviation 40 mm Hg. Systolic b...	{ "Header 1": "3.3 Normal distribution", "Header 3": "GUIDED PRACTICE 3.19", "token_count": 458, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
The normal distribution is a continuous probability distribution. Recall from Section [2.1.5](#page-96-4) that the total area under the density curve is always equal to 1, and the probability that a variable has a value within a specified interval is the area under the curve over that interval. By using either statisti...	{ "Header 1": "3.3 Normal distribution", "Header 3": "3.3.4 Calculating normal probabilities", "token_count": 1716, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Cumulative SAT scores are well-approximated by a normal model, N(1500,300). What is the probability that a randomly selected test taker scores at least 1630 on the SAT? For any normal probability problem, it can be helpful to start out by drawing the normal curve and shading the area of interest. ![](_page_156_...	{ "Header 1": "3.3 Normal distribution", "Header 3": "EXAMPLE 3.22", "token_count": 348, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Recall that the probability of a continuous random variable equaling some exact value is always 0. As a result, for a continuous random variable X, P (X ≤ x) = P (X < x) and P (X ≥ x) = P (X > x). It is valid to state that P (X ≥ x) = 1 − P (X ≤ x) = 1 − P (X < x). This is not ...	{ "Header 1": "3.3 Normal distribution", "Header 3": "DISCRETE VERSUS CONTINUOUS PROBABILITIES", "token_count": 208, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Systolic blood pressure for adults 60 years of age and older in the United States is approximately normally distributed: N(136,40). What is the probability of an adult in this age group having systolic blood pressure of 140 mm Hg or greater?[15](#page-157-1) <sup>14</sup>This probability was calculated as part of...	{ "Header 1": "3.3 Normal distribution", "Header 3": "GUIDED PRACTICE 3.24", "token_count": 226, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
The height of adult males in the United States between the ages of 20 and 62 is nearly normal, with mean 70 inches and standard deviation 3.3 inches.[16](#page-158-0) What is the probability that a random adult male is between 5'9" and 6'2"? These heights correspond to 69 inches and 74 inches. First, draw the figure....	{ "Header 1": "3.3 Normal distribution", "Header 3": "EXAMPLE 3.25", "token_count": 348, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
What percentage of adults in the United States ages 60 and older have blood pressure between 145 and 130 mm Hg?[17](#page-158-1) Z<sup>130</sup> = −0.15 → 0.4404 (area above). Final answer: 0.5890 − 0.4404 = 0.1486. <sup>16</sup>As based on a sample of 100 men, from the USDA Food Commodity Intake Databa...	{ "Header 1": "3.3 Normal distribution", "Header 3": "GUIDED PRACTICE 3.26", "token_count": 203, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
How tall is a man with height in the 40th percentile? First, draw a picture. The lower tail probability is 0.40, so the shaded area must start before the mean. ![](_page_159_Figure_4.jpeg) Determine the Z-score associated with the 40th percentile. Because the percentile is below 50%, Z will be negative. Loo...	{ "Header 1": "3.3 Normal distribution", "Header 3": "EXAMPLE 3.27", "token_count": 266, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
(a) What is the 95th percentile for SAT scores? (b) What is the 97.5 th percentile of the male heights?[18](#page-159-0) <sup>18</sup>(a) Look for 0.95 in the probability portion (middle part) of the normal probability table: row 1.6 and (about) column 0.05, i.e. Z95 = 1.65. Knowing Z95 = 1.65, µ = 15...	{ "Header 1": "3.3 Normal distribution", "Header 3": "GUIDED PRACTICE 3.28", "token_count": 222, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
The normal distribution can be used to approximate other distributions, such as the binomial distribution. The binomial formula is cumbersome when sample size is large, particularly when calculating probabilities for a large number of observations. Under certain conditions, the normal distribution can be used to approx...	{ "Header 1": "3.3 Normal distribution", "Header 3": "3.3.6 Normal approximation to the binomial distribution", "token_count": 315, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Approximately 20% of the US population smokes cigarettes. A local government commissioned a survey of 400 randomly selected individuals to investigate whether their community might have a lower smoker rate than 20%. The survey found that 59 of the 400 participants smoke cigarettes. If the true proportion of smokers in ...	{ "Header 1": "3.3 Normal distribution", "Header 3": "EXAMPLE 3.29", "token_count": 797, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
The normal model can also be used to approximate data distributions. While using a normal model can be convenient, it is important to remember that normality is always an approximation. Testing the appropriateness of the normal assumption is a key step in many data analyses. ![](_page_162_Figure_3.jpeg) Figure 3.14...	{ "Header 1": "3.3 Normal distribution", "Header 3": "3.3.7 Evaluating the normal approximation", "token_count": 308, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Three datasets were simulated from a normal distribution, with sample sizes n = 40, n = 100, and n = 400; the histograms and normal probability plots of the datasets are shown in Figure 3.15. What happens as sample size increases? As sample size increases, the data more closely follows the normal distribution; the hi...	{ "Header 1": "3.3 Normal distribution", "Header 3": "EXAMPLE 3.30", "token_count": 275, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Determine which data sets represented in Figure 3.18 plausibly come from a nearly normal distribution.<sup>21</sup> ![](_page_165_Figure_3.jpeg) Figure 3.18: Four normal probability plots for Guided Practice 3.33. (G) <sup>&</sup>lt;sup>21</sup>Answers may vary. The top-left plot shows some deviations in the sm...	{ "Header 1": "3.3 Normal distribution", "Header 3": "GUIDED PRACTICE 3.33", "token_count": 267, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
The Poisson distribution is a probability model for the number of events that occur in a population. The probability that exactly k events occur is given by $$P(X=k) = \frac{e^{-\lambda}(\lambda)^k}{k!},$$ where k may take a value 0, 1, 2, ... The mean and standard deviation of this distribution are $\lambda$ ...	{ "Header 1": "3.4 Poisson distribution", "Header 3": "POISSON DISTRIBUTION", "token_count": 379, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
For children ages 0 - 14, the incidence rate of acute lymphocytic leukemia (ALL) was approximately 30 diagnosed cases per million children per year in 2010. Approximately 20% of the US population of 319,055,000 are in this age range. What is the expected number of cases of ALL in the US over five years? The incidence...	{ "Header 1": "3.4 Poisson distribution", "Header 3": "EXAMPLE 3.37", "token_count": 268, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
(E) Recall that in the Milgram shock experiments, the probability of a person refusing to give the most severe shock is p = 0.35. Suppose that participants are tested one at a time until one person refuses; i.e., until the first occurrence of a successful trial. What are the chances that the first occurrence happens ...	{ "Header 1": "3.5 Distributions related to Bernoulli trials", "Header 3": "EXAMPLE 3.38", "token_count": 321, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
What is the probability of the first success occurring within the first 4 people? This is the probability it is the first (k = 1), second (k = 2), third (k = 3), or fourth (k = 4) trial that is the first success, which represent four disjoint outcomes. Compute the probability of each case and add the separate...	{ "Header 1": "3.5 Distributions related to Bernoulli trials", "Header 3": "EXAMPLE 3.40", "token_count": 404, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Each sequence in Figure [3.22](#page-171-0) has exactly two failures and four successes with the last attempt always being a success. If the probability of a success is p = 0.8, find the probability of the first sequence.[26](#page-171-1) If the probability of a successful extraction is p = 0.8, what is the p...	{ "Header 1": "3.5 Distributions related to Bernoulli trials", "Header 3": "GUIDED PRACTICE 3.41", "token_count": 489, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
On 70% of days, a hospital admits at least one heart attack patient. On 30% of the days, no heart attack patients are admitted. Identify each case below as a binomial or negative binomial case, and compute the probability. (a) What is the probability the hospital will admit a heart attack patient on exactly three days ...	{ "Header 1": "3.5 Distributions related to Bernoulli trials", "Header 3": "GUIDED PRACTICE 3.45", "token_count": 561, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Suppose that a large number of deer live in a forest. Researchers are interested in using the capture-recapture method to estimate total population size. A number of deer are captured in an initial sample and marked, then released; at a later time, another sample of deer are captured, and the number of marked and unmar...	{ "Header 1": "3.5 Distributions related to Bernoulli trials", "Header 3": "3.5.3 Hypergeometric distribution", "token_count": 891, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
A small clinic would like to draw a random sample of 10 individuals from their patient list of 120, of which 30 patients are smokers. (a) What is the probability of 6 individuals in the sample being smokers? (b) What is the probability that at least 2 individuals in the sample smoke?<sup>31</sup> <sup>31</sup>(a) Let...	{ "Header 1": "3.5 Distributions related to Bernoulli trials", "Header 3": "GUIDED PRACTICE 3.47", "token_count": 367, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Example [3.8](#page-145-0) calculated the variability in health care costs for an employee and her partner relying on the assumption that the number of health episodes between the two are not related. It could be reasonable to assume that the health status of one person gives no information about the other's health, gi...	{ "Header 1": "3.6 Distributions for pairs of random variables", "token_count": 491, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
The joint distribution pX,Y (x,y) for a pair of random variables (X,Y ) is the collection of probabilities $$p(x_i, y_j) = P(X = x_i \text{ and } Y = y_j)$$ for all pairs of values (x<sup>i</sup> ,yj ) that the random variables X and Y take on. Joint distributions are often displayed in tabular form as ...	{ "Header 1": "3.6 Distributions for pairs of random variables", "Header 3": "JOINT DISTRIBUTION", "token_count": 1140, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
(E) If it is known that the employee's annual health care cost is \$968, what is the conditional distribution of the partner's annual health care cost? Note that there is a different conditional distribution of Y for every possible value of X; this problem specifically asks for the conditional distribution of *...	{ "Header 1": "3.6 Distributions for pairs of random variables", "Header 3": "EXAMPLE 3.48", "token_count": 307, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Consider two random variables, X and Y, with the joint distribution shown in Figure 3.27. - (a) Compute the marginal distributions of X and Y. - (b) Identify the joint probability $p_{X,Y}(1,2)$ . - (c) What is the value of $p_{X,Y}(2,1)$ ? - (d) Compute the conditional distribution of X given that Y = 2....	{ "Header 1": "3.6 Distributions for pairs of random variables", "Header 3": "GUIDED PRACTICE 3.49", "token_count": 806, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
(E) Demonstrate that the employee's health care costs and the partner's health care costs are not independent random variables. As shown in Example 3.48, the conditional distribution of the partner's annual health care cost given that the employee's annual cost is \$968 is P(Y = \$968\|X = \$968) = 0.60, P(Y = \$988...	{ "Header 1": "3.6 Distributions for pairs of random variables", "Header 3": "EXAMPLE 3.50", "token_count": 376, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Based on Figure 3.27, check whether X and Y are independent.<sup>34</sup> Two random variables that are not independent are called correlated random variables. The correlation between two random variables is a measure of the strength of the relationship between them, just as it was for pairs of data points explor...	{ "Header 1": "3.6 Distributions for pairs of random variables", "Header 3": "GUIDED PRACTICE 3.51", "token_count": 859, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Compute the correlation between annual health care costs for the employee and her partner. As calculated previously, E(X) = \$1010, Var(X) = 1556, E(Y) = \$980, and Var(Y) = 96. Thus, SD(X) = \$39.45 and SD(Y) = \$9.80. $$\begin{split} \rho_{X,Y} &= p(x_1,y_1) \frac{(x_1 - \mu_X)}{\mathrm{sd}(X)} \frac{(y_1 - \mu_Y...	{ "Header 1": "3.6 Distributions for pairs of random variables", "Header 3": "EXAMPLE 3.55", "token_count": 547, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Based on Figure 3.27, compute the correlation between X and Y. For your convenience, the following values are provided: E(X) = 2.2, Var(X) = 2.16, E(Y) = 1.5, Var(Y) = 0.25.<sup>35</sup> When two random variables X and Y are correlated: Variance( $$X + Y$$ ) = Variance( $X$ ) + Variance( $Y$ ) + $2\sigma_X...	{ "Header 1": "3.6 Distributions for pairs of random variables", "Header 3": "GUIDED PRACTICE 3.56", "token_count": 264, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
The Association of American Medical Colleges (AAMC) introduced a new version of the Medical College Admission Test (MCAT) in the spring of 2015. Data from the scores were recently released by AAMC.[37](#page-182-0) The test consists of 4 components: chemical and physical foundations of biological systems; critical anal...	{ "Header 1": "3.6 Distributions for pairs of random variables", "Header 3": "EXAMPLE 3.61", "token_count": 529, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Thinking in terms of random variables and distributions of probabilities makes it easier to describe all possible outcomes of an experiment or process of interest, versus only considering probabilities on the scale of individual outcomes or sets of outcomes. Several of the fundamental concepts of probability can natura...	{ "Header 1": "3.7 Notes", "token_count": 313, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
[3.7](#page-442-0) Underage drinking, Part I. Data collected by the Substance Abuse and Mental Health Services Administration (SAMSHA) suggests that 69.7% of 18-20 year olds consumed alcoholic beverages in any given year.[38](#page-185-0) - (a) Suppose a random sample of ten 18-20 year olds is taken. Is the use o...	{ "Header 1": "3.8 Exercises", "Header 3": "3.8.2 Binomial distribution", "token_count": 2043, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
[3.37](#page-443-8) Computing Poisson probabilities. This is a simple exercise in computing probabilities for a Poisson random variable. Suppose that X is a Poisson random variable with rate parameter λ = 2. Calculate P (X = 2), P (X ≤ 2), and P (X ≥ 3). **[3.38](#page-443-9) Stenographer's typos....	{ "Header 1": "3.8 Exercises", "Header 3": "3.8.4 Poisson distribution", "token_count": 1077, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
[3.50](#page-444-1) Defective rate. A machine that produces a special type of transistor (a component of computers) has a 2% defective rate. The production is considered a random process where each transistor is independent of the others. - (a) What is the probability that the 10th transistor produced is the fi...	{ "Header 1": "3.8 Exercises", "Header 3": "3.8. EXERCISES 195", "token_count": 1277, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
- [4.1](#page-200-0) [Variability in estimates](#page-200-0) - [4.2](#page-204-0) [Confidence intervals](#page-204-0) - [4.3](#page-211-0) [Hypothesis testing](#page-211-0) - [4.4](#page-224-0) [Notes](#page-224-0) - [4.5](#page-226-0) [Exercises](#page-226-0) Not surprisingly, many studies are no...	{ "Header 1": "Foundations for inference", "token_count": 1637, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
How would one estimate the difference in average weight between men and women? Given that xmen = 185.1 lbs and xwomen = 162.3 lbs, what is a good point estimate for the population difference?[7](#page-200-2) Point estimates become more accurate with increasing sample size. Figure [4.3](#page-200-3) shows the ...	{ "Header 1": "4.1 Variability in estimates", "Header 3": "GUIDED PRACTICE 4.1", "token_count": 480, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
The sample mean weight calculated from cdc.samp is 173.3 lbs. Another random sample of 60 participants might produce a different value of x, such as 169.5 lbs; repeated random sampling could result in additional different values, perhaps 172.1 lbs, 168.5 lbs, and so on. Each sample mean x can be thought of as a sin...	{ "Header 1": "4.1 Variability in estimates", "Header 3": "4.1.1 The sampling distribution for the mean", "token_count": 263, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
If a sample consists of at least 30 independent observations and the data are not strongly skewed, then the distribution of the sample mean is well approximated by a normal model. ![](_page_202_Figure_1.jpeg) Figure 4.5: The left panel shows a histogram of the sample means for 100,000 random samples. The right pane...	{ "Header 1": "4.1 Variability in estimates", "Header 3": "CENTRAL LIMIT THEOREM, INFORMAL DESCRIPTION", "token_count": 223, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
The standard error (SE) of the sample mean measures the sample-to-sample variability of X, error the extent to which values of the repeated sample means oscillate around the population mean. The theoretical standard error of the sample mean is calculated by dividing the population standard deviation (σ<sup>x</sup> ...	{ "Header 1": "4.1 Variability in estimates", "Header 3": "4.1.2 Standard error of the mean", "token_count": 482, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
- The population mean and standard deviation are denoted by µ and σ. - The sample mean and standard deviation are denoted by x and s. - The distribution of the random variable X refers to the collection of sample means if multiple samples of the same size were repeatedly drawn from a population. - The mean of...	{ "Header 1": "4.1 Variability in estimates", "Header 3": "SUMMARY: POINT ESTIMATE TERMINOLOGY", "token_count": 232, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
While a point estimate consists of a single value, an interval estimate provides a plausible range of values for a parameter. When estimating a population mean µ, a confidence interval for µ has the general form $$(\overline{x}-m, \overline{x}+m) = \overline{x} \pm m,$$ where m is the margin of error. Interva...	{ "Header 1": "4.2 Confidence intervals", "Header 3": "4.2.1 Interval estimates for a population parameter", "token_count": 766, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
The sample mean adult weight from the 60 observations in cdc.samp is xweight = 173.3 lbs, and the standard deviation is sweight = 49.04 lbs. Use Equation [4.2](#page-204-2) to calculate an approximate 95% confidence interval for the average adult weight in the US population. The standard error for the sample ...	{ "Header 1": "4.2 Confidence intervals", "Header 3": "EXAMPLE 4.3", "token_count": 345, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Let Y be a normally distributed random variable. Ninety-nine percent of the time, Y will be within how many standard deviations of the mean? This is equivalent to the z-score with 0.005 area to the right of z and 0.005 to the left of −z. In the normal probability table, this is the z-value that with 0.005...	{ "Header 1": "4.2 Confidence intervals", "Header 3": "EXAMPLE 4.6", "token_count": 585, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Create a 99% confidence interval for the average adult weight in the US population using the data in cdc.samp. The point estimate is xweight = 173.3 and the standard error is SE<sup>x</sup> = 6.33. Apply the 99% confidence interval formula: xweight ± 2.58×SE<sup>x</sup> → (156.97,189.63). A data a...	{ "Header 1": "4.2 Confidence intervals", "Header 3": "EXAMPLE 4.8", "token_count": 605, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
One-sided confidence intervals for a population mean provide either a lower bound or an upper bound, but not both. One-sided confidence intervals have the form $(\overline{x} - m, \infty)$ or $(-\infty, \overline{x} + m)$ . While the margin of error m for a one-sided interval is still calculated from the standa...	{ "Header 1": "4.2 Confidence intervals", "Header 3": "4.2.3 One-sided confidence intervals", "token_count": 268, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
The correct interpretation of an XX% confidence interval is, "We are XX% confident that the population parameter is between ..." While it may be tempting to say that a confidence interval captures the population parameter with a certain probability, this is a common error. The confidence level only quantifies how plaus...	{ "Header 1": "4.2 Confidence intervals", "Header 3": "4.2.4 Interpreting confidence intervals", "token_count": 457, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Body mass index (BMI) is one measure of body weight that adjusts for height. The National Health and Nutrition Examination Survey (NHANES) consists of a set of surveys and measurements conducted by the US CDC to assess the health and nutritional status of adults and children in the United States. The dataset nhanes.sam...	{ "Header 1": "4.2 Confidence intervals", "Header 3": "EXAMPLE 4.11", "token_count": 813, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Important decisions in science, such as whether a new treatment for a disease should be approved for the market, are primarily data-driven. For example, does a clinical study of a new cholesterol-lowering drug provide robust evidence of a beneficial effect in patients at risk for heart disease? A confidence interval ca...	{ "Header 1": "4.3 Hypothesis testing", "token_count": 239, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
The null hypothesis (H0) often represents either a skeptical perspective or a claim to be tested. The alternative hypothesis (HA) is an alternative claim and is often represented by a range of possible parameter values. Generally, an investigator suspects that the null hypothesis is not true and performs a hypoth...	{ "Header 1": "4.3 Hypothesis testing", "Header 3": "NULL AND ALTERNATIVE HYPOTHESES", "token_count": 218, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
The claim to be tested is that the population average of the difference between actual and desired weight for US adults is equal to 0. $$H_0: \mu = 0.$$ In the absence of prior evidence that people typically wish to be lighter (or heavier), it is reasonable to begin with an alternative hypothesis that allows for di...	{ "Header 1": "4.3 Hypothesis testing", "Header 3": "Step 1: Formulating null and alternative hypotheses", "token_count": 354, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Calculating the test statistic t is analogous to standardizing observations with Z-scores as discussed in Chapter 3. The test statistic quantifies the number of standard deviations between the sample mean x and the population mean µ: $$t = \frac{\overline{x} - \mu_0}{s/\sqrt{n}},$$ where s is the sample sta...	{ "Header 1": "4.3 Hypothesis testing", "Header 3": "Step 3: Calculating the test statistic", "token_count": 230, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
The p-value is the probability of observing a sample mean as or more extreme than the observed value, under the assumption that the null hypothesis is true. In samples of size 40 or more, the t-statistic will have a standard normal distribution unless the data are strongly skewed or extreme outliers are present. Re...	{ "Header 1": "4.3 Hypothesis testing", "Header 3": "Step 4: Calculating the p-value", "token_count": 556, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
To reach a conclusion about the null hypothesis, directly compare p and α. Note that for a conclusion to be informative, it must be presented in the context of the original question; it is not useful to only state whether or not H<sup>0</sup> is rejected. If p > α, the observed sample mean is not extreme enou...	{ "Header 1": "4.3 Hypothesis testing", "Header 3": "Step 5: Drawing a conclusion", "token_count": 270, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Suppose that the mean weight difference in the sampled group of 60 adults had been 7 pounds instead of 18.2 pounds, but with the same standard deviation of 33.46 pounds. Would there still be enough evidence at the α = 0.05 level to reject H<sup>0</sup> : µ = 0 in favor of H<sup>A</sup> : µ , 0?[15](#page-21...	{ "Header 1": "4.3 Hypothesis testing", "Header 3": "GUIDED PRACTICE 4.12", "token_count": 267, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
While fish and other types of seafood are important for a healthy diet, nearly all fish and shellfish contain traces of mercury. Dietary exposure to mercury can be particularly dangerous for young children and unborn babies. Regulatory organizations such as the US Food and Drug Administration (FDA) provide guidelines a...	{ "Header 1": "4.3 Hypothesis testing", "Header 3": "EXAMPLE 4.13", "token_count": 1060, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
In 2015, the National Sleep Foundation published new guidelines for the amount of sleep recommended for adults: 7-9 hours of sleep per night.[18](#page-216-1) The NHANES survey includes a question asking respondents about how many hours per night they sleep; the responses are available in nhanes.samp. In the sample of ...	{ "Header 1": "4.3 Hypothesis testing", "Header 3": "EXAMPLE 4.14", "token_count": 609, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
The relationship between a hypothesis test and the corresponding confidence interval is defined by the significance level α; the two approaches are based on the same inferential logic, and differ only in perspective. The hypothesis testing approach asks whether x is far enough away from µ<sup>0</sup> to be consid...	{ "Header 1": "4.3 Hypothesis testing", "Header 3": "4.3.3 Hypothesis testing and confidence intervals", "token_count": 666, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Calculate the confidence interval for the average mercury level for bluefin tuna caught off the coast of New Jersey. The summary statistics for the sample of 21 fish are $\bar{x} = 0.53$ ppm and s = 0.16 ppm. Does the interval agree with the results of Example 4.13? The 95% confidence interval is: $$\overline{x} ...	{ "Header 1": "4.3 Hypothesis testing", "Header 3": "EXAMPLE 4.16", "token_count": 593, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Previously, a hypothesis test was conducted at $\alpha = 0.05$ to test the null hypothesis $H_0: \mu = 7$ hours against the alternative $H_A: \mu < 7$ hours, for the average sleep per night US adults. Calculate the corresponding one-sided confidence interval and compare the information obtained from a confidence ...	{ "Header 1": "4.3 Hypothesis testing", "Header 3": "EXAMPLE 4.17", "token_count": 504, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Hypothesis tests can potentially result in incorrect decisions, such as rejecting the null hypothesis when the null is actually true. Figure 4.17 shows the four possible ways that the conclusion of a test can be right or wrong. \| \| \| Test conclusion \| ...	{ "Header 1": "4.3 Hypothesis testing", "Header 3": "4.3.4 Decision errors", "token_count": 276, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
In a trial, the defendant is either innocent (H0) or guilty (HA). After hearing evidence from both the prosecution and the defense, the court must reach a verdict. What does a Type I Error represent in this context? What does a Type II Error represent? If the court makes a Type I error, this means the defendant i...	{ "Header 1": "4.3 Hypothesis testing", "Header 3": "EXAMPLE 4.18", "token_count": 300, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Reducing the error probability of one type of error increases the chance of making the other type. As a result, the significance level is often adjusted based on the consequences of any decisions that might follow from the result of a significance test. By convention, most scientific studies use a significance level ...	{ "Header 1": "4.3 Hypothesis testing", "Header 3": "Choosing a significance level", "token_count": 560, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
In some cases, the choice of a one-sided or two-sided test can influence whether the null hypothesis is rejected. For example, consider a sample for which the t-statistic is 1.80. If a twosided test is conducted at α = 0.05, the p-value is $$P(Z \le -\|t\|) + P(Z \ge \|t\|) = 2P(Z \ge 1.80) = 0.072.$$ There is ...	{ "Header 1": "4.3 Hypothesis testing", "Header 3": "4.3.5 Choosing between one-sided and two-sided tests", "token_count": 1166, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Formal hypothesis tests are designed for settings where a decision or a claim about a hypothesis follows a test, such as in scientific publications where an investigator wishes to claim that an intervention changes an outcome. However, progress in science is usually based on a collection of studies or experiments, and ...	{ "Header 1": "4.3 Hypothesis testing", "Header 3": "4.3.6 The informal use of p-values", "token_count": 286, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Confidence intervals and hypothesis testing are two of the central concepts in inference for a population based on a sample. The confidence interval shows a range of population parameter values consistent with the observed sample, and is often used to design additional studies. Hypothesis testing is a useful tool for e...	{ "Header 1": "4.4 Notes", "token_count": 874, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
[4.4](#page-445-4) Mental health, Part I. The 2010 General Social Survey asked the question: "For how many days during the past 30 days was your mental health, which includes stress, depression, and problems with emotions, not good?" Based on responses from 1,151 US residents, the survey reported a 95% confidence i...	{ "Header 1": "4.5 Exercises", "Header 3": "4.5.2 Confidence intervals", "token_count": 1259, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
[4.8](#page-446-1) Age at first marriage, Part I. The National Survey of Family Growth conducted by the Centers for Disease Control gathers information on family life, marriage and divorce, pregnancy, infertility, use of contraception, and men's and women's health. One of the variables collected on this survey is t...	{ "Header 1": "4.5 Exercises", "Header 3": "4.5. EXERCISES 231", "token_count": 1184, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
[4.13](#page-446-6) Online communication. A study suggests that the average college student spends 10 hours per week communicating with others online. You believe that this is an underestimate and decide to collect your own sample for a hypothesis test. You randomly sample 60 students from your dorm and find that o...	{ "Header 1": "4.5 Exercises", "Header 3": "4.5. EXERCISES 233", "token_count": 1612, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
[4.21](#page-447-2) Testing for fibromyalgia. A patient named Diana was diagnosed with fibromyalgia, a long-term syndrome of body pain, and was prescribed anti-depressants. Being the skeptic that she is, Diana didn't initially believe that anti-depressants would help her symptoms. However after a couple months of b...	{ "Header 1": "4.5 Exercises", "Header 3": "4.5. EXERCISES 235", "token_count": 805, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
The tools studied in Chapter [4](#page-197-0) all made use of the t-statistic from a sample mean, $$t = \frac{\overline{x} - \mu}{s/\sqrt{n}},$$ where the parameter µ is a population mean, x and s are the sample mean and standard deviation, and n is the sample size. Tests and confidence intervals were res...	{ "Header 1": "5.1 Single-sample inference with the t-distribution", "token_count": 280, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }
Figure [5.1](#page-237-1) shows a t-distribution and normal distribution. Like the standard normal distribution, the t-distribution is unimodal and symmetric about zero. However, the tails of a t-distribution are thicker than for the normal, so observations are more likely to fall beyond two standard deviations f...	{ "Header 1": "5.1 Single-sample inference with the t-distribution", "Header 3": "5.1.1 The t-distribution", "token_count": 431, "source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf" }