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In many instances, we are given a conditional probability of the form
P(statement about variable 1 | statement about variable 2)
but we would really like to know the inverted conditional probability:
P(statement about variable 2 | statement about variable 1)
Tree diagrams can be used to find the second conditio... | {
"Header 1": "2.2.7 Bayes' Theorem",
"token_count": 2036,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
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If Jose comes to campus and finds the garage full, what is the probability that there is a sporting event? Use a tree diagram to solve this problem.<sup>40</sup>
• Example 2.58 Here we solve the same problem presented in Exercise 2.57, except this time we use Bayes' Theorem.
The outcome of interest is whether there... | {
"Header 1": "2.2.7 Bayes' Theorem",
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Example 2.61 Professors sometimes select a student at random to answer a question. If each student has an equal chance of being selected and there are 15 people in your class, what is the chance that she will pick you for the next question?
If there are 15 people to ask and none are skipping class, then the probabili... | {
"Header 1": "2.3 Sampling from a small population (special topic)",
"token_count": 1365,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
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Example 2.68 Two books are assigned for a statistics class: a textbook and its corresponding study guide. The university bookstore determined 20% of enrolled students do not buy either book, 55% buy the textbook, and 25% buy both books, and these percentages are relatively constant from one term to another. If there ar... | {
"Header 1": "2.4 Random variables (special topic)",
"token_count": 462,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
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We call a variable or process with a numerical outcome a random variable, and we usually represent this random variable with a capital letter such as X, Y , or Z. The amount of money a single student will spend on her statistics books is a random variable, and we represent it by X.
```
Random variable
```
A random ... | {
"Header 1": "2.4.1 Expectation",
"token_count": 1107,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
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Suppose you ran the university bookstore. Besides how much revenue you expect to generate, you might also want to know the volatility (variability) in your revenue.
The variance and standard deviation can be used to describe the variability of a random variable. Section [1.6.4](#page-34-0) introduced a method for fin... | {
"Header 1": "2.4.2 Variability in random variables",
"token_count": 2020,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
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If X represents the profit for selling the TV and Y represents the cost of the toaster oven, write an equation that represents the net change in Elena's cash.[50](#page-109-0)
- J Exercise 2.79 Based on past auctions, Elena figures she should expect to make about \$175 on the TV and pay about \$23 for the toaster oven.... | {
"Header 1": "2.4.2 Variability in random variables",
"token_count": 1100,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
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Quantifying the average outcome from a linear combination of random variables is helpful, but it is also important to have some sense of the uncertainty associated with the total outcome of that combination of random variables. The expected net gain or loss of Leonard's stock portfolio was considered in Exercise [2.83.... | {
"Header 1": "2.4.4 Variability in linear combinations of random variables",
"token_count": 1610,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
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Example 2.88 Figure [2.26](#page-113-1) shows a few different hollow histograms of the variable height for 3 million US adults from the mid-90's.[58](#page-113-2) How does changing the number of bins allow you to make different interpretations of the data?
Adding more bins provides greater detail. This sample is extr... | {
"Header 1": "2.5 Continuous distributions (special topic)",
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"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
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Examine the transition from a boxy hollow histogram in the top-left of Figure [2.26](#page-113-1) to the much smoother plot in the lower-right. In this last plot, the bins are so slim that the hollow histogram is starting to resemble a smooth curve. This suggests the population height as a continuous numerical variable... | {
"Header 1": "2.5.1 From histograms to continuous distributions",
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"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
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We computed the proportion of individuals with heights 180 to 185 cm in Example [2.89](#page-113-3) as a fraction:
> number of people between 180 and 185 total sample size
We found the number of people with heights between 180 and 185 cm by determining the fraction of the histogram's area in this region. Similarly,... | {
"Header 1": "2.5.2 Probabilities from continuous distributions",
"token_count": 600,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
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Data collected at elementary schools in DeKalb County, GA suggest that each year roughly 25% of students miss exactly one day of school, 15% miss 2 days, and 28% miss 3 or more days due to sickness.[65](#page-118-1)
- (a) What is the probability that a student chosen at random doesn't miss any days of school due to s... | {
"Header 1": "2.6.1 Defining probability",
"token_count": 1150,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
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2.15 Joint and conditional probabilities. P(A) = 0.3, P(B) = 0.7
- (a) Can you compute P(A and B) if you only know P(A) and P(B)?
- (b) Assuming that events A and B arise from independent random processes,
- i. what is P(A and B)?
- ii. what is P(A or B)?
- iii. what is P(A|B)?
- (c) If we are given that P(A and B) =... | {
"Header 1": "2.6.2 Conditional probability",
"token_count": 2039,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
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The normal distribution model always describes a symmetric, unimodal, bell shaped curve. However, these curves can look different depending on the details of the model. Specifically, the normal distribution model can be adjusted using two parameters: mean and standard deviation. As you can probably guess, changing the ... | {
"Header 1": "3.1 Normal distribution",
"Header 3": "3.1.1 Normal distribution model",
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"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
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Example 3.2 Table [3.4](#page-129-1) shows the mean and standard deviation for total scores on the SAT and ACT. The distribution of SAT and ACT scores are both nearly normal. Suppose Ann scored 1800 on her SAT and Tom scored 24 on his ACT. Who performed better?
We use the standard deviation as a guide. Ann is 1 stand... | {
"Header 1": "3.1.2 Standardizing with Z scores",
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"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
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• Example 3.7 Ann from Example 3.2 earned a score of 1800 on her SAT with a corresponding Z = 1. She would like to know what percentile she falls in among all SAT test-takers.
Ann's **percentile** is the percentage of people who earned a lower SAT score than Ann. We shade the area representing those individuals in Fi... | {
"Header 1": "3.1.2 Standardizing with Z scores",
"Header 3": "3.1.3 Normal probability table",
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"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
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Cumulative SAT scores are approximated well by a normal model, N(µ = 1500, σ = 300).
Example 3.9 Shannon is a randomly selected SAT taker, and nothing is known about Shannon's SAT aptitude. What is the probability Shannon scores at least 1630 on her SATs?
First, always draw and label a picture of the normal distrib... | {
"Header 1": "3.1.4 Normal probability examples",
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"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
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Identifying the mean µ = 1500, the standard deviation σ = 300, and the cutoff for the tail area x = 1400 makes it easy to compute the Z score:
$$Z = \frac{x - \mu}{\sigma} = \frac{1400 - 1500}{300} = -0.33$$
Using the normal probability table, identify the row of −0.3 and column of ... | {
"Header 1": "3.1.4 Normal probability examples",
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"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
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then we can find the middle area:
That is, the probability of being between 5'9'' and 6'2'' is 0.5048.
$\odot$ Exercise 3.20 What percent of SAT takens get between 1500 and 2000?<sup>15</sup>
$\bigcirc$ Exercise 3.21 What percent of adult males are between 5'5" and 5'7"?<sup>16</... | {
"Header 1": "3.1.4 Normal probability examples",
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"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
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<sup>&</sup>lt;sup>15</sup>This is an abbreviated solution. (Be sure to draw a figure!) First find the percent who get below 1500 and the percent that get above 2000: $Z_{1500} = 0.00 \rightarrow 0.5000$ (area below), $Z_{2000} = 1.67 \rightarrow 0.0475$ (area above). Final answer: 1.0000 - 0.5000 - 0.0475 = 0.45... | {
"Header 1": "3.1.4 Normal probability examples",
"token_count": 232,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
} |
Here, we present a useful rule of thumb for the probability of falling within 1, 2, and 3 standard deviations of the mean in the normal distribution. This will be useful in a wide range of practical settings, especially when trying to make a quick estimate without a calculator or Z table.
... | {
"Header 1": "3.1.5 68-95-99.7 rule",
"token_count": 416,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
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Many processes can be well approximated by the normal distribution. We have already seen two good examples: SAT scores and the heights of US adult males. While using a normal model can be extremely convenient and helpful, it is important to remember normality is
<sup>17</sup>First draw the pictures. To find the area ... | {
"Header 1": "3.2 Evaluating the normal approximation",
"token_count": 364,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
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Example 3.15 suggests the distribution of heights of US males is well approximated by the normal model. We are interested in proceeding under the assumption that the data are normally distributed, but first we must check to see if this is reasonable.
There are two visual methods for checking the assumption of normali... | {
"Header 1": "3.2 Evaluating the normal approximation",
"Header 3": "**3.2.1** Normal probability plot",
"token_count": 1459,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
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We construct a normal probability plot for the heights of a sample of 100 men as follows:
- (1) Order the observations.
- (2) Determine the percentile of each observation in the ordered data set.
- (3) Identify the Z score corresponding to each percentile.
- (4) Create a scatterplot of the observations (vertical) aga... | {
"Header 1": "3.2 Evaluating the normal approximation",
"Header 3": "3.2.2 Constructing a normal probability plot (special topic)",
"token_count": 430,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
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Stanley Milgram began a series of experiments in 1963 to estimate what proportion of people would willingly obey an authority and give severe shocks to a stranger. Milgram found that about 65% of people would obey the authority and give such shocks. Over the years, additional research suggested this number is approxima... | {
"Header 1": "3.3.1 Bernoulli distribution",
"token_count": 774,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
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Example 3.29 Dr. Smith wants to repeat Milgram's experiments but she only wants to sample people until she finds someone who will not inflict the worst shock.[25](#page-143-2) If the probability a person will not give the most severe shock is still 0.35 and the subjects are independent, what are the chances that she wi... | {
"Header 1": "3.3.2 Geometric distribution",
"token_count": 1969,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
} |
Example 3.37 Suppose we randomly selected four individuals to participate in the "shock" study. What is the chance exactly one of them will be a success? Let's call the four people Allen (A), Brittany (B), Caroline (C), and Damian (D) for convenience. Also, suppose 35% of people are successes as in the previous version... | {
"Header 1": "3.4 Binomial distribution (special topic)",
"token_count": 421,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
} |
The scenario outlined in Example [3.37](#page-146-3) is a special case of what is called the binomial distribution. The binomial distribution describes the probability of having exactly k successes in n independent Bernoulli trials with probability of a success p (in Example [3.37,](#page-146-3) n = 4, k = 1, p = 0.35)... | {
"Header 1": "3.4.1 The binomial distribution",
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"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
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Is the binomial model appropriate? What is the probability that (a) none of them will develop a severe lung condition? (b) One will develop a severe lung condition? (c) That no more than one will develop a severe lung condition?<sup>33</sup>
- Exercise 3.46 What is the probability that at least 2 of your 4 smoking frie... | {
"Header 1": "3.4.1 The binomial distribution",
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"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
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The binomial formula is cumbersome when the sample size (n) is large, particularly when we consider a range of observations. In some cases we may use the normal distribution as an easier and faster way to estimate binomial probabilities.
Example 3.50 Approximately 20% of the US population smokes cigarettes. A local g... | {
"Header 1": "3.4.2 Normal approximation to the binomial distribution",
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"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
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Caution: The normal approximation may fail on small intervals The normal approximation to the binomial distribution tends to perform poorly when estimating the probability of a small range of counts, even when the conditions are met.
Suppose we wanted to compute the probability of observing 69, 70, or 71 smokers in 4... | {
"Header 1": "3.4.3 The normal approximation breaks down on small intervals",
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"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
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The geometric distribution describes the probability of observing the first success on the n th trial. The negative binomial distribution is more general: it describes the probability of observing the k th success on the n th trial.
Example 3.54 Each day a high school football coach tells his star kicker, Brian, that... | {
"Header 1": "3.5.1 Negative binomial distribution",
"token_count": 2007,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
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$$\binom{n-1}{k-1}p^k(1-p)^{n-k} = \frac{5!}{3!2!}(0.8)^4(0.2)^2 = 10 \times 0.0164 = 0.164$$
- J Exercise 3.60 The negative binomial distribution requires that each kick attempt by Brian is independent. Do you think it is reasonable to suggest that each of Brian's kick attempts are independent?[43](#page-155-1)
- ... | {
"Header 1": "3.5.1 Negative binomial distribution",
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Example 3.63 There are about 8 million individuals in New York City. How many individuals might we expect to be hospitalized for acute myocardial infarction (AMI), i.e. a heart attack, each day? According to historical records, the average number is about 4.4 individuals. However, we would also like to know the approxi... | {
"Header 1": "3.5.2 Poisson distribution",
"token_count": 1099,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
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The distribution of passenger vehicle speeds traveling on the Interstate 5 Freeway (I-5) in California is nearly normal with a mean of 72.6 miles/hour and a standard deviation of 4.78 miles/hour.[47](#page-160-0)
- (a) What percent of passenger vehicles travel slower than 80 miles/hour?
- (b) What percent of passenge... | {
"Header 1": "3.6.1 Normal distribution",
"token_count": 1094,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
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3.27 Underage drinking, Part I. The Substance Abuse and Mental Health Services Administration estimated that 70% of 18-20 year olds consumed alcoholic beverages in 2008.[49](#page-163-1)
- (a) Suppose a random sample of ten 18-20 year olds is taken. Is the use of the binomial distribution appropriate for calculating ... | {
"Header 1": "3.6.4 Binomial distribution",
"token_count": 2033,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
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Statistical inference is concerned primarily with understanding the quality of parameter estimates. For example, a classic inferential question is, "How sure are we that the estimated mean, $\bar{x}$ , is near the true population mean, $\mu$ ?" While the equations and details change depending on the setting, the foun... | {
"Header 1": "Foundations for inference",
"token_count": 996,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
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We want to estimate the population mean based on the sample. The most intuitive way to go about doing this is to simply take the sample mean. That is, to estimate the average 10 mile run time of all participants, take the average time for the sample:
$$\bar{x} = \frac{88.22 + 100.58 + \dots + 89.40}{100} = 95.61$$
... | {
"Header 1": "4.1.1 Point estimates",
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"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
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Estimates are usually not exactly equal to the truth, but they get better as more data become available. We can see this by plotting a running mean from our run10Samp sample. A running mean is a sequence of means, where each mean uses one more observation in its calculation than the mean directly before it in the seque... | {
"Header 1": "4.1.2 Point estimates are not exact",
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"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
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From the random sample represented in run10Samp, we guessed the average time it takes to run 10 miles is 95.61 minutes. Suppose we take another random sample of 100 individuals and take its mean: 95.30 minutes. Suppose we took another (93.43 minutes) and another (94.16 minutes), and so on. If we do this many many times... | {
"Header 1": "4.1.3 Standard error of the mean",
"token_count": 1354,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
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We achieved three goals in this section. First, we determined that point estimates from a sample may be used to estimate population parameters. We also determined that these point estimates are not exact: they vary from one sample to another. Lastly, we quantified the uncertainty of the sample mean using what we call t... | {
"Header 1": "4.1.3 Standard error of the mean",
"Header 3": "4.1.4 Basic properties of point estimates",
"token_count": 448,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
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Our point estimate is the most plausible value of the parameter, so it makes sense to build the confidence interval around the point estimate. The standard error, which is a measure of the uncertainty associated with the point estimate, provides a guide for how large we should make the confidence interval.
The standa... | {
"Header 1": "4.2.2 An approximate 95% confidence interval",
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"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
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In Section [4.1.3,](#page-171-0) we introduced a sampling distribution for ¯x, the average run time for samples of size 100. We examined this distribution earlier in Figure [4.7.](#page-172-0) Now we'll take 100,000 samples, calculate the mean of each, and plot them in a histogram to get an especially accurate depictio... | {
"Header 1": "4.2.3 A sampling distribution for the mean",
"token_count": 509,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
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Suppose we want to consider confidence intervals where the confidence level is somewhat higher than 95%: perhaps we would like a confidence level of 99%. Think back to the analogy about trying to catch a fish: if we want to be more sure that we will catch the fish, we should use a wider net. To create a 99% confidence ... | {
"Header 1": "4.2.4 Changing the confidence level",
"token_count": 1443,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
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In rare circumstances we know important characteristics of a population. For instance, we might know a population is nearly normal and we may also know its parameter values. Even so, we may still like to study characteristics of a random sample from the population. Consider the conditions required for modeling a sample... | {
"Header 1": "4.2.6 Nearly normal population with known SD (special topic)",
"token_count": 1232,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
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The average time for all runners who finished the Cherry Blossom Run in 2006 was 93.29 minutes (93 minutes and about 17 seconds). We want to determine if the run10Samp data set provides strong evidence that the participants in 2012 were faster or slower than those runners in 2006, versus the other possibility that ther... | {
"Header 1": "4.3.1 Hypothesis testing framework",
"token_count": 888,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
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We can start the evaluation of the hypothesis setup by comparing 2006 and 2012 run times using a point estimate from the 2012 sample: ¯x<sup>12</sup> = 95.61 minutes. This estimate suggests the average time is actually longer than the 2006 time, 93.29 minutes. However, to evaluate whether this provides strong evidence ... | {
"Header 1": "4.3.2 Testing hypotheses using confidence intervals",
"token_count": 1531,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
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Hypothesis tests are not flawless. Just think of the court system: innocent people are sometimes wrongly convicted and the guilty sometimes walk free. Similarly, we can make a wrong decision in statistical hypothesis tests. However, the difference is that we have the tools necessary to quantify how often we make such e... | {
"Header 1": "4.3.3 Decision errors",
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"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
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The p-value is a way of quantifying the strength of the evidence against the null hypothesis and in favor of the alternative. Formally the p-value is a conditional probability.
#### p-value
The p-value is the probability of observing data at least as favorable to the alternative hypothesis as our current data set, ... | {
"Header 1": "4.3.4 Formal testing using p-values",
"token_count": 1994,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
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J Exercise 4.30 If the null hypothesis is true, how often should the p-value be less than 0.05?[26](#page-189-0)
<sup>26</sup>About 5% of the time. If the null hypothesis is true, then the data only has a 5% chance of being in the 5% of data most favorable to HA.
Figure 4.17: A hist... | {
"Header 1": "4.3.4 Formal testing using p-values",
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We now consider how to compute a p-value for a two-sided test. In one-sided tests, we shade the single tail in the direction of the alternative hypothesis. For example, when the alternative had the form $\mu > 7$ , then the p-value was represented by the upper tail (Figure 4.16). When the alternative was $\mu < 46.99... | {
"Header 1": "4.3.4 Formal testing using p-values",
"Header 3": "4.3.5 Two-sided hypothesis testing with p-values",
"token_count": 1295,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
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Choosing a significance level for a test is important in many contexts, and the traditional level is 0.05. However, it is often helpful to adjust the significance level based on the application. We may select a level that is smaller or larger than 0.05 depending on the consequences of any conclusions reached from the t... | {
"Header 1": "4.3.6 Choosing a significance level",
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"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
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The Central Limit Theorem states that when the sample size is small, the normal approximation may not be very good. However, as the sample size becomes large, the normal approximation improves. We will investigate three cases to see roughly when the approximation is reasonable.
We consider three data sets: one from a... | {
"Header 1": "Central Limit Theorem, informal definition The distribution of ¯x is approximately normal. The approximation can be poor if the sample size is small, but it improves with larger sample sizes.",
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"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
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The sample mean is not the only point estimate for which the sampling distribution is nearly normal. For example, the sampling distribution of sample proportions closely resembles the normal distribution when the sample size is sufficiently large. In this section, we introduce a number of examples where the normal appr... | {
"Header 1": "4.5 Inference for other estimators",
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"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
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In Section [4.2,](#page-174-0) we used the point estimate ¯x with a standard error SEx¯ to create a 95% confidence interval for the population mean:
$$\bar{x} \pm 1.96 \times SE_{\bar{x}} \tag{4.44}$$
We constructed this interval by noting that the sample mean is within 1.96 standard errors of the actual mean about... | {
"Header 1": "4.5.1 Confidence intervals for nearly normal point estimates",
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"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
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Just as the confidence interval method works with many other point estimates, we can generalize our hypothesis testing methods to new point estimates. Here we only consider the p-value approach, introduced in Section [4.3.4,](#page-186-0) since it is the most commonly used technique and also extends to non-normal cases... | {
"Header 1": "4.5.2 Hypothesis testing for nearly normal point estimates",
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"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
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Statistical tools rely on conditions. When the conditions are not met, these tools are unreliable and drawing conclusions from them is treacherous. The conditions for these tools typically come in two forms.
• The individual observations must be independent. A random sample from less than 10% of the population ensure... | {
"Header 1": "4.5.4 When to retreat",
"token_count": 319,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
} |
Many companies are concerned about rising healthcare costs. A company may estimate certain health characteristics of its employees, such as blood pressure, to project its future cost obligations. However, it might be too expensive to measure the blood pressure of every employee at a large company, and the company may c... | {
"Header 1": "4.6.1 Finding a sample size for a certain margin of error",
"token_count": 790,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
} |
Consider the following two hypotheses:
- H0: The average blood pressure of employees is the same as the national average, µ = 130.
- HA: The average blood pressure of employees is different than the national average, µ 6= 130.
Suppose the alternative hypothesis is actually true. Then we might like to know, what is ... | {
"Header 1": "4.6.2 Power and the Type 2 Error rate",
"token_count": 1288,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
} |
When the sample size becomes larger, point estimates become more precise and any real differences in the mean and null value become easier to detect and recognize. Even a very small difference would likely be detected if we took a large enough sample. Sometimes researchers will take such large samples that even the sli... | {
"Header 1": "4.6.3 Statistical significance versus practical significance",
"token_count": 360,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
} |
4.1 Identify the parameter, Part I. For each of the following situations, state whether the parameter of interest is a mean or a proportion. It may be helpful to examine whether individual responses are numerical or categorical.
- (a) In a survey, one hundred college students are asked how many hours per week they sp... | {
"Header 1": "4.7.1 Variability in estimates",
"token_count": 1330,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
} |
4.7 Relaxing after work. The General Social Survey (GSS) is a sociological survey used to collect data on demographic characteristics and attitudes of residents of the United States. In 2010, the survey collected responses from 1,154 US residents. The survey is conducted face-to-face with an in-person interview of a ra... | {
"Header 1": "4.7.2 Confidence intervals",
"token_count": 1800,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
} |
4.15 Identify hypotheses, Part I. Write the null and alternative hypotheses in words and then symbols for each of the following situations.
- (a) New York is known as "the city that never sleeps". A random sample of 25 New Yorkers were asked how much sleep they get per night. Do these data provide convincing evidence... | {
"Header 1": "4.7.3 Hypothesis testing",
"token_count": 2038,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
} |
Is there evidence that the nutrition label does not provide an accurate measure of calories in the bags of potato chips? We have verified the independence, sample size, and skew conditions are satisfied.
4.26 Find the sample mean. You are given the following hypotheses: H0: µ = 34, HA: µ > 34. We know that the sample... | {
"Header 1": "4.7.3 Hypothesis testing",
"token_count": 1339,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
} |
4.33 Ages of pennies, Part I. The histogram below shows the distribution of ages of pennies at a bank.
- (a) Describe the distribution.
- (b) Sampling distributions for means from simple random samples of 5, 30, and 100 pennies is shown in the histograms below. Describe the shapes of these distributions and comment o... | {
"Header 1": "4.7.4 Examining the Central Limit Theorem",
"token_count": 1748,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
} |
4.43 Spam mail, Part I. The 2004 National Technology Readiness Survey sponsored by the Smith School of Business at the University of Maryland surveyed 418 randomly sampled Americans, asking them how many spam emails they receive per day. The survey was repeated on a new random sample of 499 Americans in 2009.[47](#page... | {
"Header 1": "4.7.5 Inference for other estimators",
"token_count": 847,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
} |
Are textbooks actually cheaper online? Here we compare the price of textbooks at UCLA's bookstore and prices at Amazon.com. Seventy-three UCLA courses were randomly sampled in Spring 2010, representing less than 10% of all UCLA courses.<sup>1</sup> A portion of this data set is shown in Table 5.1
| | dept | cou... | {
"Header 1": "5.1 Paired data",
"token_count": 426,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
} |
Each textbook has two corresponding prices in the data set: one for the UCLA bookstore and one for Amazon. Therefore, each textbook price from the UCLA bookstore has a natural correspondence with a textbook price from Amazon. When two sets of observations have this special correspondence, they are said to be paired.
... | {
"Header 1": "5.1.1 Paired observations and samples",
"token_count": 305,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
} |
To analyze a paired data set, we use the exact same tools that we developed in Chapter [4.](#page-168-0) Now we apply them to the differences in the paired observations.
<sup>2</sup>Observation 2: 40.59 − 31.14 = 9.45. Observation 3: 31.68 − 32.00 = −0.32.
| ndiff | x¯diff | sdiff |
|-------|--------|-------|
| 73 ... | {
"Header 1": "5.1.2 Inference for paired data",
"token_count": 898,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
} |
We would like to estimate the average difference in run times for men and women using the run10Samp data set, which was a simple random sample of 45 men and 55 women from all runners in the 2012 Cherry Blossom Run. Table [5.5](#page-224-3) presents relevant summary statistics, and box plots of each sample are shown in ... | {
"Header 1": "5.2.1 Point estimates and standard errors for differences of means",
"token_count": 1101,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
} |
When the data indicate that the point estimate $\bar{x}_1 - \bar{x}_2$ comes from a nearly normal distribution, we can construct a confidence interval for the difference in two means from the framework built in Chapter 4. Here a point estimate, $\bar{x}_w - \bar{x}_m = 14.48$ , is associated with a normal model with... | {
"Header 1": "5.2.1 Point estimates and standard errors for differences of means",
"Header 3": "5.2.2 Confidence interval for the difference",
"token_count": 295,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
} |
A data set called baby\_smoke represents a random sample of 150 cases of mothers and their newborns in North Carolina over a year. Four cases from this data set are represented in Table 5.7. We are particularly interested in two variables: weight and smoke. The weight variable represents the weights of the newborns and... | {
"Header 1": "5.2.1 Point estimates and standard errors for differences of means",
"Header 3": "5.2.3 Hypothesis tests based on a difference in means",
"token_count": 1864,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
} |
When considering the difference of two means, there are two common cases: the two samples are paired or they are independent. (There are instances where the data are neither paired nor independent.) The paired case was treated in Section [5.1,](#page-221-1) where the one-sample methods were applied to the differences f... | {
"Header 1": "5.2.4 Summary for inference of the difference of two means",
"token_count": 889,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
} |
The motivation in Chapter 4 for requiring a large sample was two-fold. First, a large sample ensures that the sampling distribution of $\bar{x}$ is nearly normal. We will see in Section 5.3.1 that if the population data are nearly normal, then $\bar{x}$ is also nearly normal regardless of the
<sup>&</sup>lt;sup>1... | {
"Header 1": "5.3 One-sample means with the t distribution",
"token_count": 437,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
} |
The second reason we previously required a large sample size was so that we could accurately estimate the standard error using the sample data. In the cases where we will use a small sample to calculate the standard error, it will be useful to rely on a new distribution for inference calculations: the t distribution. A... | {
"Header 1": "5.3.2 Introducing the t distribution",
"token_count": 2030,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
} |
When estimating the mean and standard error from a small sample, the t distribution is a more accurate tool than the normal model. This is true for both small and large samples.
#### TIP: When to use the t distribution
Use the t distribution for inference of the sample mean when observations are independent and nea... | {
"Header 1": "5.3.3 The t distribution as a solution to the standard error problem",
"token_count": 407,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
} |
Dolphins are at the top of the oceanic food chain, which causes dangerous substances such as mercury to concentrate in their organs and muscles. This is an important problem for both dolphins and other animals, like humans, who occasionally eat them. For instance, this is particularly relevant in Japan where school mea... | {
"Header 1": "5.3.3 The t distribution as a solution to the standard error problem",
"Header 3": "5.3.4 One sample t confidence intervals",
"token_count": 1501,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
} |
An SAT preparation company claims that its students' scores improve by over 100 points on average after their course. A consumer group would like to evaluate this claim, and they collect data on a random sample of 30 students who took the class. Each of these students took the SAT before and after taking the company's ... | {
"Header 1": "5.3.3 The t distribution as a solution to the standard error problem",
"Header 3": "5.3.5 One sample t tests",
"token_count": 1560,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
} |
In the example of two exam versions, the teacher would like to evaluate whether there is convincing evidence that the difference in average scores between the two exams is not due to chance.
It will be useful to extend the t distribution method from Section 5.3 to apply to a difference of means:
$$\bar{x}_1 - \bar{... | {
"Header 1": "5.4 The *t* distribution for the difference of two means",
"Header 3": "5.4.1 Sampling distributions for the difference in two means",
"token_count": 671,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
} |
Summary statistics for each exam version are shown in Table [5.19.](#page-240-2) The teacher would like to evaluate whether this difference is so large that it provides convincing evidence that Version B was more difficult (on average) than Version A.
| Version | n | x<br>¯ | s | min | max |
|---------|----|-------... | {
"Header 1": "5.4.2 Two sample t test",
"token_count": 2033,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
} |
Occasionally, two populations will have standard deviations that are so similar that they can be treated as identical. For example, historical data or a well-understood biological mechanism may justify this strong assumption. In such cases, we can make our t distribution approach slightly more precise by using a pooled... | {
"Header 1": "5.4.3 Two sample t confidence interval",
"Header 3": "5.4.4 Pooled standard deviation estimate (special topic)",
"token_count": 631,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
} |
Sometimes we want to compare means across many groups. We might initially think to do pairwise comparisons; for example, if there were three groups, we might be tempted to compare the first mean with the second, then with the third, and then finally compare the second and third means for a total of three comparisons. H... | {
"Header 1": "5.5 Comparing many means with ANOVA (special topic)",
"token_count": 781,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
} |
We would like to discern whether there are real differences between the batting performance of baseball players according to their position: outfielder (OF), infielder (IF), designated hitter (DH), and catcher (C). We will use a data set called bat10, which includes batting records of 327 Major League Baseball (MLB) pl... | {
"Header 1": "5.5.1 Is batting performance related to player position in MLB?",
"token_count": 1685,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
} |
The method of analysis of variance in this context focuses on answering one question: is the variability in the sample means so large that it seems unlikely to be from chance alone? This question is different from earlier testing procedures since we will simultaneously consider many groups, and evaluate whether their s... | {
"Header 1": "5.5.2 Analysis of variance (ANOVA) and the F test",
"token_count": 1585,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
} |
The calculations required to perform an ANOVA by hand are tedious and prone to human error. For these reasons, it is common to use statistical software to calculate the F statistic and p-value.
An ANOVA can be summarized in a table very similar to that of a regression summary, which we will see in Chapters [7](#page-... | {
"Header 1": "5.5.3 Reading an ANOVA table from software",
"token_count": 320,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
} |
There are three conditions we must check for an ANOVA analysis: all observations must be independent, the data in each group must be nearly normal, and the variance within each group must be approximately equal.
- Independence. If the data are a simple random sample from less than 10% of the population, this conditio... | {
"Header 1": "5.5.4 Graphical diagnostics for an ANOVA analysis",
"token_count": 598,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
} |
When we reject the null hypothesis in an ANOVA analysis, we might wonder, which of these groups have different means? To answer this question, we compare the means of each possible pair of groups. For instance, if there are three groups and there is strong evidence that there are some differences in the group means, th... | {
"Header 1": "5.5.5 Multiple comparisons and controlling Type 1 Error rate",
"token_count": 2034,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
} |
5.1 Global warming, Part I. Is there strong evidence of global warming? Let's consider a small scale example, comparing how temperatures have changed in the US from 1968 to 2008. The daily high temperature reading on January 1 was collected in 1968 and 2008 for 51 randomly selected locations in the continental US. Then... | {
"Header 1": "5.6.1 Paired data",
"token_count": 1306,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
} |
5.7 Math scores of 13 year olds, Part I. The National Assessment of Educational Progress tested a simple random sample of 1,000 thirteen year old students in both 2004 and 2008 (two separate simple random samples). The average and standard deviation in 2004 were 257 and 39, respectively. In 2008, the average and standa... | {
"Header 1": "5.6.2 Difference of two means",
"token_count": 1773,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
} |
5.23 Cleveland vs. Sacramento. Average income varies from one region of the country to another, and it often reflects both lifestyles and regional living expenses. Suppose a new graduate is considering a job in two locations, Cleveland, OH and Sacramento, CA, and he wants to see whether the average income in one of the... | {
"Header 1": "5.6.4 The t distribution for the difference of two means",
"token_count": 2007,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
} |
5.37 Chicken diet and weight, Part III. In Exercises [5.29](#page-265-2) and [5.31](#page-265-4) we compared the effects of two types of feed at a time. A better analysis would first consider all feed types at once: casein, horsebean, linseed, meat meal, soybean, and sunflower. The ANOVA output below can be used to tes... | {
"Header 1": "5.6.5 Comparing many means with ANOVA",
"token_count": 2044,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
} |
A sample proportion can be described as a sample mean. If we represent each "success" as a 1 and each "failure" as a 0, then the sample proportion is the mean of these numerical outcomes:
$$\hat{p} = \frac{0+1+1+\dots+0}{976} = 0.44$$
The distribution of ˆp is nearly normal when the distribution of 0's and 1's is n... | {
"Header 1": "6.1.1 Identifying when the sample proportion is nearly normal",
"token_count": 571,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
} |
We may want a confidence interval for the proportion of Americans who approve of the job the Supreme Court is doing. Our point estimate, based on a sample of size n = 976 from the NYTimes/CBS poll, is ˆp = 0.44. To use the general confidence interval formula from Section [4.5,](#page-197-0) we must check the conditions... | {
"Header 1": "6.1.2 Confidence intervals for a proportion",
"token_count": 705,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
} |
To apply the normal distribution framework in the context of a hypothesis test for a proportion, the independence and success-failure conditions must be satisfied. In a hypothesis test, the success-failure condition is checked using the null proportion: we verify np<sup>0</sup> and n(1 − p0) are at least 10, where p<su... | {
"Header 1": "6.1.3 Hypothesis testing for a proportion",
"token_count": 737,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
} |
We first encountered sample size computations in Section [4.6,](#page-202-0) which considered the case of estimating a single mean. We found that these computations were helpful in planning a study to control the size of the standard error of a point estimate. The task was to find a sample size n so that the sample mea... | {
"Header 1": "6.1.4 Choosing a sample size when estimating a proportion",
"token_count": 1225,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
} |
We must check two conditions before applying the normal model to $\hat{p}_1 - \hat{p}_2$ . First, the sampling distribution for each sample proportion must be nearly normal, and secondly, the samples must be independent. Under these two conditions, the sampling distribution of $\hat{p}_1 - \hat{p}_2$ may be well app... | {
"Header 1": "6.2 Difference of two proportions",
"Header 3": "6.2.1 Sample distribution of the difference of two proportions",
"token_count": 737,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
} |
In the setting of confidence intervals, the sample proportions are used to verify the successfailure condition and also compute standard error, just as was the case with a single proportion.
Example 6.10 The way a question is phrased can influence a person's response. For example, Pew Research Center conducted a surv... | {
"Header 1": "6.2.2 Intervals and tests for p<sup>1</sup> − p<sup>2</sup>",
"token_count": 1431,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
} |
Here we use a new example to examine a special estimate of standard error when H<sup>0</sup> : p<sup>1</sup> = p2. We investigate whether there is an increased risk of cancer in dogs that are exposed to the herbicide 2,4-dichlorophenoxyacetic acid (2,4-D). A study in 1994 examined 491 dogs that had developed cancer and... | {
"Header 1": "6.2.3 Hypothesis testing when H<sup>0</sup> : p<sup>1</sup> = p<sup>2</sup>",
"token_count": 1672,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
} |
In this section, we develop a method for assessing a null model when the data are binned. This technique is commonly used in two circumstances:
- Given a sample of cases that can be classified into several groups, determine if the sample is representative of the general population.
- Evaluate whether data resemble a ... | {
"Header 1": "6.3 Testing for goodness of fit using chi-square (special topic)",
"token_count": 626,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
} |
Example 6.18 Of the people in the city, 275 served on a jury. If the individuals are randomly selected to serve on a jury, about how many of the 275 people would we expect to be white? How many would we expect to be black?
About 72% of the population is white, so we would expect about 72% of the jurors to be white: 0... | {
"Header 1": "6.3.1 Creating a test statistic for one-way tables",
"token_count": 516,
"source_pdf": "datasets/websources/Med_v1/med_textbook/OpenIntroStatSecond.pdf"
} |
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