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)2(0.2) = 1.352 2 4 6 8 0.1 0.3 0.4 0.2 Table 4.24 21. Identify the mistake in the probability distribution table. x P(x) x*P(x) 1 2 3 4 5 0.15 0.15 0.25 0.50 0.30 0.90 0.20 0.80 0.15 0.75 Table 4.25 22. Identify the mistake in the probability distribution table. x P(x) x*P(x) 1 2 3 4 5 0.15 0.15 0.25 0.40 0.25 0.65 0.... |
_______ 31. P(x < 4) = _______ 32. On average, how many years would you expect a child to study ballet with this teacher? 33. What does the column "P(x)" sum to and why? 34. What does the column "x*P(x)" sum to and why? 35. You are playing a game by drawing a card from a standard deck and replacing it. If the card is a... |
: The Higher Education Research Institute at UCLA collected data from 203,967 incoming first-time, full-time freshmen from 270 four-year colleges and universities in the U.S. 71.3% of those students replied that, yes, they believe that same-sex couples should have the right to legal marital status. Suppose that you ran... |
can the Poisson model be used to approximate? When would you do this? Use the following information to answer the next six exercises: On average, eight teens in the U.S. die from motor vehicle injuries per day. As a result, states across the country are debating raising the driving age. 63. Assume the event occurs ind... |
coin and is equally likely to land on heads or tails. • • • If the card is a face card, and the coin lands on Heads, you win $6 If the card is a face card, and the coin lands on Tails, you win $2 If the card is not a face card, you lose $2, no matter what the coin shows. a. Find the expected value for this game (expec... |
chance of losing the million dollars. The third company, a biotech firm, has a 10% chance of returning $6,000,000 profit, a 70% of no profit or loss, and a 20% chance of losing the million dollars. a. Construct a PDF for each investment. b. Find the expected value for each investment. c. Which is the safest investment... |
three DVDs. c. Find the probability that a customer rents at least four DVDs. d. Find the probability that a customer rents at most two DVDs. Another shop, Entertainment Headquarters, rents DVDs and video games. The probability distribution for DVD rentals per customer at this shop is given as follows. They also have ... |
VARIABLES c. Find the mean of X. d. Find the standard deviation of X. 81. In a lottery, there are 250 prizes of $5, 50 prizes of $25, and ten prizes of $100. Assuming that 10,000 tickets are to be issued and sold, what is a fair price to charge to break even? 4.3 Binomial Distribution 82. According to a recent article... |
the values that X may take on. c. Give the distribution of X. X ~ _____(_____,_____) d. How many of the 12 students do we expect to attend the festivities? e. Find the probability that at most four students will attend. f. Find the probability that more than two students will attend. Use the following information to a... |
four dice will show a one? Use numbers to justify your answer numerically. In words, define the random variable X. 95. More than 96 percent of the very largest colleges and universities (more than 15,000 total enrollments) have some online offerings. Suppose you randomly pick 13 such institutions. We are interested in... |
TE RANDOM VARIABLES c. Give the distribution of X. X ~ _____(_____,_____) d. How many seniors are expected to have participated in after-school sports all four years of high school? e. Based on numerical values, would you be surprised if none of the seniors participated in after-school sports all four years of high sch... |
. If none of the dice show the number or object that was bet, the house keeps the $1 bet. If one of the dice shows the number or object bet (and the other two do not show it), the player gets back his or her $1 bet, plus $1 profit. If two of the dice show the number or object bet (and the third die does not show it), t... |
104. A consumer looking to buy a used red Miata car will call dealerships until she finds a dealership that carries the car. She estimates the probability that any independent dealership will have the car will be 28%. We are interested in the number of dealerships she must call. In words, define the random variable X.... |
? g. How many California residents do you expect to need to survey until you find a California resident who does have adequate earthquake supplies? 107. In one of its Spring catalogs, L.L. Bean® advertised footwear on 29 of its 192 catalog pages. Suppose we randomly survey 20 pages. We are interested in the number of p... |
are infected with HIV.”[1] In South Africa, the prevalence of HIV is 17.3%. Let X = the number of people you test until you find a person infected with HIV. 1. ”Prevalence of HIV, total (% of populations ages 15-49),” The World Bank, 2013. Available online at http://data.worldbank.org/indicator/ 274 CHAPTER 4 | DISCRE... |
values that X may take on. c. Give the distribution of X. X ~ _____(_____,_____) d. How many pages do you expect to advertise footwear on them? e. Calculate the standard deviation. 114. Suppose that a technology task force is being formed to study technology awareness among instructors. Assume that ten people will be ... |
quity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 4 | DISCRETE RANDOM VARIABLES 275 117. The switchboard in a Minneapolis law office gets an average of 5.5 incoming phone calls during the noon hour on Mondays. Experience shows that the existing staff can handle up to six calls in an h... |
. c. Give the distribution of X. X ~ _____(_____,_____) d. Find the probability that she has no children. e. Find the probability that she has fewer children than the Spanish average. f. Find the probability that she has more children than the Spanish average. 122. Fertile, female cats produce an average of three litte... |
2% per year. Suppose that 100 people with tax returns over $25,000 are randomly picked. We are interested in the number of people audited in one year. Use a Poisson distribution to anwer the following questions. In words, define the random variable X. a. b. List the values that X may take on. c. Give the distribution ... |
Mrs. Plum’s cats wake her up at night because they want to play is ten. We are interested in the number of times her cats wake her up each week. 128. In words, the random variable X = _________________ a. b. c. d. the number of times Mrs. Plum’s cats wake her up each week. the number of times Mrs. Plum’s cats wake her... |
: Few demographic differences seen in these views other than by income,” GALLUP® Economy, 2013. Available online at http://www.gallup.com/poll/162368/americansenjoy-saving-rather-spending.aspx (accessed May 15, 2013). Pryor, John H., Linda DeAngelo, Laura Palucki Blake, Sylvia Hurtado, Serge Tran. The American Freshman... |
Angelo, Laura Palucki Blake, Sylvia Hurtado, Serge Tran. The American Freshman: National Norms Fall 2011. Los Angeles: Cooperative Institutional Research Program at the Higher Education Research Institute at http://heri.ucla.edu/PDFs/pubs/TFS/Norms/Monographs/ at Also TheAmericanFreshman2011.pdf (accessed May 15, 2013)... |
state.sc.us/ dmh/anorexia/statistics.htm (accessed May 15, 2013). “Giving Birth in Manila: The maternity ward at the Dr Jose Fabella Memorial Hospital in Manila, the busiest in the Philippines, where at http://www.theguardian.com/world/gallery/2011/jun/08/philippines-health#/?picture=375471900&index=2 (accessed May 15,... |
10 + 0.05 = 0.15 5 1 7 0.35 + 0.40 + 0.10 = 0.85 9 1(0.15) + 2(0.35) + 3(0.40) + 4(0.10) = 0.15 + 0.70 + 1.20 + 0.40 = 2.45 11 x P(x) 0 1 2 3 0.03 0.04 0.08 0.85 Table 4.39 13 Let X = the number of events Javier volunteers for each month. 15 This content is available for free at http://textbookequity.org/introductory-s... |
yes" that same-sex couples should have the right to legal marital status. 47 1,2,… 49 1.4 51 X = the number of business majors in the sample. 53 2, 3, 4, 5, 6, 7, 8, 9 55 6.26 57 0, 1, 2, 3, 4, … 59 0.0485 61 0.0214 280 CHAPTER 4 | DISCRETE RANDOM VARIABLES 63 X = the number of U.S. teens who die from motor vehicle inj... |
. 1.6 This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 4 | DISCRETE RANDOM VARIABLES 281 75 a. Software Company x 5,000,000 1,000,000 –1,000,000 Table 4.42 P(x) 0.10 0.30 0.60 Hardware Company x 3,000,000 1,000,000 –1,000,00 Table... |
than or equal to 24." • Using your calculator's distribution menu: 1 – binomcdf ⎛ ⎝32, 1 3, 24 ⎞ ⎠ • P(x > 24) = 0 • The probability of getting more than 75% of the 32 questions correct when randomly guessing is very small and practically zero. 95 a. X = the number of college and universities that offer online offerin... |
TE RANDOM VARIABLES c. P(x > 5) = 1 – P(x ≤ 5) = 1 – binomcdf(15, 0.281, 5) = 1 – 0.7754 = 0.2246 P(x = 3) = binompdf(15, 0.281, 3) = 0.1927 P(x = 4) = binompdf(15, 0.281, 4) = 0.2259 It is more likely that four people are literate that three people are. 105 a. X = the number of adults in America who are surveyed until... |
introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 4 | DISCRETE RANDOM VARIABLES 285 d. Without replacement 117 a. X ~ P(5.5); μ = 5.5; σ = 5.5 ≈ 2.3452 b. P(x ≤ 6) = poissoncdf(5.5, 6) ≈ 0.6860 c. There is a 15.7% probability that the law staff will receive more calls than they can handle. d. P... |
5 286 CHAPTER 4 | DISCRETE RANDOM VARIABLES e. 0.2231 f. 0.0001 g. Yes 129 d This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 5 | CONTINUOUS RANDOM VARIABLES 287 5 | CONTINUOUS RANDOM VARIABLES Figure 5.1 The heights of these radi... |
The entire area under the curve and above the x-axis is equal to one. • Probability is found for intervals of x values rather than for individual x values. • P(c < x < d) is the probability that the random variable X is in the interval between the values c and d. P(c < x < d) is the area under the curve, above the x-a... |
to represent the probability that the value of the random variable X is in the interval between one and two. 5.1 | Continuous Probability Functions We begin by defining a continuous probability density function. We use the function notation f(x). Intermediate algebra may have been your first formal introduction to fun... |
RANDOM VARIABLES 291 Figure 5.7 ⎛ AREA = (15 – 4) ⎝ ⎛ AREA = (15 – 4) ⎝ ⎞ ⎠ = 0.55 ⎞ ⎠ = 0.55 1 20 1 20 (15 – 4) = 11 = the base of a rectangle The area corresponds to the probability P(4 < x < 15) = 0.55. Suppose we want to find P(x = 15). On an x-y graph, x = 15 is a vertical line. A vertical line has no width (or z... |
careful to note if the data is inclusive or exclusive. Example 5.2 The data in Table 5.1 are 55 smiling times, in seconds, of an eight-week-old baby. This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 5 | CONTINUOUS RANDOM VARIABLE... |
the theoretical mean and standard deviation are close to the sample mean and standard deviation in this example. 5.2 The data that follow are the number of passengers on 35 different charter fishing boats. The sample mean = 7.9 and the sample standard deviation = 4.33. The data follow a uniform distribution where all ... |
you already know the baby has smiled for more than eight seconds. Find P(x > 12|x > 8) There are two ways to do the problem. For the first way, use the fact that this is a conditional and changes the sample space. The graph illustrates the new sample space. You already know the baby smiled more than eight seconds. Wri... |
deviation, σ. Solution 5.4 b. μ = a + b 2 = 15 + 0 2 σ = (b - a)2 12 = (15 - 0)2 12 = 7.5. On the average, a person must wait 7.5 minutes. = 4.3. The Standard deviation is 4.3 minutes. c. Ninety percent of the time, the time a person must wait falls below what value? NOTE This asks for the 90th percentile. Solution 5.... |
ut for more than 1.5 minutes. 298 CHAPTER 5 | CONTINUOUS RANDOM VARIABLES The second question has a conditional probability. You are asked to find the probability that a nine-year old child eats a donut in more than two minutes given that the child has already been eating the donut for more than 1.5 minutes. Solve the ... |
a furnace. Then x ~ U (1.5, 4). This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 5 | CONTINUOUS RANDOM VARIABLES 299 a. Find the probability that a randomly selected furnace repair requires more than two hours. b. Find the probab... |
.75 = k – 1.5, obtained by dividing both sides by 0.4 k = 2.25, obtained by adding 1.5 to both sides The 30th percentile of repair times is 2.25 hours. 30% of repair times are 2.5 hours or less. Solution 5.6 d. Figure 5.21 Uniform Distribution between 1.5 and 4 with an area of 0.25 shaded to the right representing the ... |
telephone calls, and the amount of time, in months, a car battery lasts. It can be shown, too, that the value of the change that you have in your pocket or purse approximately follows an exponential distribution. Values for an exponential random variable occur in the following way. There are fewer large values and mor... |
and graph the distribution. Example 5.8 a. Using the information in Exercise 5.0, find the probability that a clerk spends four to five minutes with a randomly selected customer. Solution 5.8 a. Find P(4 < x < 5). The cumulative distribution function (CDF) gives the area to the left. P(x < x) = 1 – e–mx P(x < 5) = 1 –... |
50) -0.25 following two notes. NOTE A formula for the percentile k is k = ln(1 − AreaToTheLe f t) −m where ln is the natural log. 304 CHAPTER 5 | CONTINUOUS RANDOM VARIABLES On the home screen, enter ln(1 – 0.50)/–0.25. Press the (-) for the negative. c. Which is larger, the mean or the median? Solution 5.8 c. From par... |
> 7) = e(–0.1)(7) = 0.4966. The probability that a computer part lasts more than seven years is 0.4966. This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 5 | CONTINUOUS RANDOM VARIABLES 305 On the home screen, enter e^(-.1*7). Fig... |
Eighty percent of running shoes last at most how long if used every day? Example 5.10 Suppose that the length of a phone call, in minutes, is an exponential random variable with decay parameter =. If another person arrives at a public telephone just before you, find the probability that you will have to wait 1 12 more... |
∼ Exp(0.5). The cumulative distribution function is P(X < x) = 1 – e(–0.5x)e. Therefore P(X < 1) = 1 – e(–0.5)(1) ≈ 0.3935. 1 - e^(–0.5) ≈ 0.3935 308 CHAPTER 5 | CONTINUOUS RANDOM VARIABLES Figure 5.28 d. P(X > 5) = 1 – P(X < 5) = 1 – (1 – e(–5)(0.5)) = e–2.5 ≈ 0.0821. Figure 5.29 1 – (1 – e^( – 5*0.5)) or e^( – 5*0.5... |
. On average, how many seconds elapse between two successive cars? b. After a car passes by, how long on average will it take for another seven cars to pass by? c. Find the probability that after a car passes by, the next car will pass within the next 20 seconds. d. Find the probability that after a car passes by, the ... |
| CONTINUOUS RANDOM VARIABLES Example 5.12 Refer to Example 5.7 where the time a postal clerk spends with his or her customer has an exponential distribution with a mean of four minutes. Suppose a customer has spent four minutes with a postal clerk. What is the probability that he or she will spend at least an additio... |
org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 5 | CONTINUOUS RANDOM VARIABLES 311 Suppose X has the Poisson distribution with mean λ. Compute P(X = k) by entering 2nd, VARS(DISTR), C: poissonpdf(λ, k). To compute P(X ≤ k), enter 2nd, VARS (DISTR), D:poissoncdf(λ, k). Example 5.13 At a p... |
) = 45 e−4 5! ≈ 0.1563. (5! = (5)(4)(3)(2)(1)) 312 CHAPTER 5 | CONTINUOUS RANDOM VARIABLES poissonpdf(4, 5) = 0.1563. d. Keep in mind that X must be a whole number, so P(X < 5) = P(X ≤ 4). To compute this, we could take P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4). Using technology, we see that P(X ≤ 4) = 0.628... |
__________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ ... |
are they? Either way, justify your answer numerically. (Recall that any DATA that are less than Q1 – 1.5(IQR) or more than Q3 + 1.5(IQR) are potential outliers. IQR means interquartile range.) Compare the Data 1. For each of the following parts, use a complete sentence to comment on how the value obtained from the dat... |
X > x + k|X > x) = P(X > k). the memoryless property is the statement Poisson distribution If there is a known average of λ events occurring per unit time, and these events are independent of each other, then the number of events X occurring in one unit of time has the Poisson distribution. The probability of k events ... |
b). All values x are equally likely. We write X ∼ U(a, b). The mean of X is µ =. The standard deviation of X is a + b 2 CHAPTER 5 | CONTINUOUS RANDOM VARIABLES 317 (b − a)2 σ = 12 of X is P(X ≤ x is continuous.. The probability density function of X is f (x) = 1 for a ≤ x ≤ b. The cumulative distribution function b − ... |
X ≤ x) 5.2 The Uniform Distribution X = a real number between a and b (in some instances, X can take on the values a and b). a = smallest X; b = largest X X ~ U (a, b) The mean is µ = a + b 2 Area to the Left of x: P(X < x) = (x – a) ⎛ ⎝ 1 b − a ⎞ ⎠ Area to the Right of x: P(X > x) = (b – x) ⎛ ⎝ 1 b − a ⎞ ⎠ Area Betwee... |
illustrate? Figure 5.37 2. Which type of distribution does the graph illustrate? Figure 5.38 3. Which type of distribution does the graph illustrate? This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 5 | CONTINUOUS RANDOM VARIABLE... |
3.8 2.5 1.5 Table 5.4 This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 5 | CONTINUOUS RANDOM VARIABLES 321 2.8 1.8 4.5 1.9 1.9 3.1 1.6 Table 5.4 The sample mean = 2.50 and the sample standard deviation = 0.8302. The distribution ... |
VARIABLES Figure 5.45 Identify the following values: b. i. Lowest value for x¯ : _______ ii. Highest value for x¯ : _______ iii. Height of the rectangle: _______ iv. Label for x-axis (words): _______ v. Label for y-axis (words): _______ 41. Find the average age of the cars in the lot. 42. Find the probability that a r... |
6). 55. Find the 70th percentile. 324 CHAPTER 5 | CONTINUOUS RANDOM VARIABLES Use the following information to answer the next seven exercises. A distribution is given as X ~ Exp(0.75). 56. What is m? 57. What is the probability density function? 58. What is the cumulative distribution function? 59. Draw the distribut... |
percent of nurses in America with an R.N. (registered nurse) degree. You ask nurses if they have an R.N. degree. The nurses answer “yes” or “no.” You then calculate the percentage of nurses with an R.N. degree. You give that percentage to your supervisor. a. What part of the experiment will yield discrete data? b. Wha... |
_________ a. Define the random variable. X = _________ b. X ~ _________ c. Graph the probability distribution. d. e. μ = _________ f. σ = _________ g. Find the probability that the individual lost more than ten pounds in a month. h. Suppose it is known that the individual lost more than ten pounds in a month. Find the... |
. four 80. Find the 30th percentile for the waiting times (in minutes). a. two b. 2.4 c. 2.75 d. three 81. The probability of waiting more than seven minutes given a person has waited more than four minutes is? a. 0.125 b. 0.25 c. 0.5 d. 0.75 This content is available for free at http://textbookequity.org/introductory-... |
find the probability that the stock is more than $21. 84. A fireworks show is designed so that the time between fireworks is between one and five seconds, and follows a uniform distribution. a. Find the average time between fireworks. b. Find probability that the time between fireworks is greater than four seconds. 85... |
discrete? 328 CHAPTER 5 | CONTINUOUS RANDOM VARIABLES d. μ = ________ e. σ = ________ f. Draw a graph of the probability distribution. Label the axes. g. Find the probability that the percent is less than 12. h. Find the probability that the percent is between eight and 14. i. The percent of all individuals living in ... |
a. 0.1941 b. 1.3863 c. 2.0794 d. 5.5452 94. Let X ~ Exp(0.1). a. decay rate = ________ b. μ = ________ c. Graph the probability distribution function. d. On the graph, shade the area corresponding to P(x < 6) and find the probability. e. Sketch a new graph, shade the area corresponding to P(3 < x < 6) and find the pro... |
't give up any hits throughout the game. No-hitters occur at a rate of about three per season. Assume that the duration of time between no-hitters is exponential. a. What is the probability that an entire season elapses with a single no-hitter? b. If an entire season elapses without any no-hitters, what is the probabil... |
time between two successive visits to the urgent care facility is less than 2 minutes. b. Find the probability that the time between two successive visits to the urgent care facility is more than 15 minutes. c. If 10 minutes have passed since the last arrival, what is the probability that the next person will arrive w... |
5 43 a. Check student’s solution. b. 3.5 7 45 a. Check student's solution. b. k = 7.25 c. 7.25 47 No, outcomes are not equally likely. In this distribution, more people require a little bit of time, and fewer people require a lot of time, so it is more likely that someone will require less time. 49 five 51 f(x) = 0.2e... |
(0.25)(9) = 2.25. Thus, the value is 25 – 2.25 = 22.75. d. This is a conditional probability question. P(x > 21| x > 18). You can do this two ways: ◦ Draw the graph where a is now 18 and b is still 25. The height is 1 (25 − 18) = 1 7 So, P(x > 21|x > 18) = (25 – 21) ⎛ ⎝ = 4/7. ⎞ ⎠ 1 7 ◦ Use the formula: P(x > 21|x > 1... |
(t < 6) * 10⎞ – 1 8 – 1 8 * 6⎞ = = ⎛ ⎜1 – e ⎝ ⎛ ⎜.7135 – 0.5276 = 0.1859 ⎠ Figure 5.56 This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 5 | CONTINUOUS RANDOM VARIABLES 335 c. We want to find 0.70 = P(T > t) = 1 – ⎛ ⎜. ⎞ ⎟ = e ⎠ So... |
hitters per season is Poisson with mean λ = 3. Therefore, (X = 0) = 30 e – 3 = e–3 ≈ 0.0498 0! You could let T = duration of time between no-hitters. Since the time is exponential and there are 3 no-hitters per season, then the time between no-hitters is 1 3 season. For the exponential, µ = 1 3. Therefore, m = 1 µ = 3 ... |
B people is discrete instead of continuous.) 101 Let T = duration (in minutes) between successive visits. Since patients arrive at a rate of one patient every seven minutes, μ = 7 and the decay constant is m = 1 7. The cdf is P(T < t) = 1 − e t 7 a. P(T < 2.2485. b. P(T > 15) = 1 − P(T < 15) = 1 − c. P(T > 15|T > 10) ... |
the real world. In this chapter, you will study the normal distribution, the standard normal distribution, and applications associated with them. 338 CHAPTER 6 | THE NORMAL DISTRIBUTION The normal distribution has two parameters (two numerical descriptive measures), the mean (μ) and the standard deviation (σ). If X is... |
if the mean of a normal distribution is five and the standard deviation is two, the value 11 is three standard deviations above (or to the right of) the mean. The calculation is as follows: x = μ + (z)(σ) = 5 + (3)(2) = 11 The z-score is three. The mean for the standard normal distribution is zero, and the standard de... |
, x is to the left of or below μ. Or, when z is positive, x is greater than μ, and when z is negative x is less than μ. 6.1 What is the z-score of x, when x = 1 and X ~ N(12,3)? Example 6.2 Some doctors believe that a person can lose five pounds, on the average, in a month by reducing his or her fat intake and by exerc... |
) measures the same weight gain for a second group of people. A negative weight gain would be a weight loss. Since x = 17 and y = 4 are each two standard deviations to the right of their means, they represent the same, standardized weight gain relative to their means. 6.2 Fill in the blanks. Jerome averages 16 points a... |
-old male from Chile was 168 cm tall from 2009 to 2010. The z-score when x = 168 cm is z = _______. This z-score tells you that x = 168 is ________ standard deviations to the ________ (right or left) of the mean _____ (What is the mean?). Solution 6.3 a. –0.32, 0.32, left, 170 b. Suppose that the height of a 15 to 18-y... |
162.85 cm. Interpret each z-score. What can you say about x = 160.58 cm and y = 162.85 cm? Solution 6.4 The z-score for x = 160.58 is z = –1.5. The z-score for y = 162.85 is z = –1.5. Both x = 160.58 and y = 162.85 deviate the same number of standard deviations from their respective means and in the same direction. 6.... |
to 1985, the mean height of 15 to 18-year-old males from Chile was 172.36 cm, and the standard deviation was 6.34 cm. Let Y = the height of 15 to 18-year-old males in 1984 to 1985. Then Y ~ N(172.36, 6.34). a. About 68% of the y values lie between what two values? These values are ________________. The z-scores are __... |
1 – P(X < x). Remember, P(X < x) = Area to the left of the vertical line through x. P(X < x) = 1 – P(X < x) = Area to the right of the vertical line through x. P(X < x) is the same as P(X ≤ x) and P(X > x) is the same as P(X ≥ x) for continuous distributions. Calculations of Probabilities Probabilities are calculated ... |
of the normal curve. HISTORICAL NOTE The TI probability program calculates a z-score and then the probability from the z-score. Before technology, the z-score was looked up in a standard normal probability table (because the math involved is too cumbersome) to find the probability. In this example, a standard normal t... |
lower than k, and ten percent are the same or higher. The variable k is often called a critical value. k = 69.4 Figure 6.6 346 CHAPTER 6 | THE NORMAL DISTRIBUTION The 90th percentile is 69.4. This means that 90% of the test scores fall at or below 69.4 and 10% fall at or above. To get this answer on the calculator, fo... |
/col11562/1.16 CHAPTER 6 | THE NORMAL DISTRIBUTION 347 Figure 6.7 normalcdf(1.8,2.75,2,0.5) = 0.5886 The probability that a household personal computer is used between 1.8 and 2.75 hours per day for entertainment is 0.5886. b. Find the maximum number of hours per day that the bottom quartile of households uses a person... |
the 30th percentile, and interpret it in a complete sentence. b. What is the probability that the age of a randomly selected smartphone user in the range 13 to 55+ is less than 27 years old. Example 6.11 There are approximately one billion smartphone users in the world today. In the United States the ages 13 to 55+ of... |
of 0.24 cm. a. Find the probability that a randomly selected mandarin orange from this farm has a diameter larger than 6.0 cm. Sketch the graph. Solution 6.12 a. normalcdf(6,10^99,5.85,0.24) = 0.2660 Figure 6.9 b. The middle 20% of mandarin oranges from this farm have diameters between ______ and ______. Solution 6.12... |
_______ _______ _______ _______ _______ _______ _______ _______ _______ _______ _______ _______ _______ _______ _______ _______ _______ _______ _______ _______ _______ _______ _______ _______ _______ _______ Table 6.1 2. Construct a histogram. Make five to six intervals. Sketch the graph using a ruler and pencil. Scale... |
bookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 6 | THE NORMAL DISTRIBUTION 353 6.2 Normal Distribution (Pinkie Length) Class Time: Names: Student Learning Outcomes • The student will compare empirical data and a theoretical distribution to determine if data from the experiment... |
. • The 85th percentile is _______. • Median is _______. • What is the theoretical probability that a randomly chosen pinky length is more than 6.5 cm? • Explain the meaning of the 85th percentile of this data. Discussion Questions Do the data you collected give a close approximation to the theoretical distribution? In... |
is a continuous distribution, the total area under the curve is one. The parameters of the normal are the mean µ and the standard deviation σ. A special normal distribution, called the standard normal distribution is the distribution of z-scores. Its mean is zero, and its standard deviation is one. FORMULA REVIEW 6.0 ... |
of x = 12, if it is two standard deviations to the right of the mean? 12. What is the z-score of x = 9, if it is 1.5 standard deviations to the left of the mean? 13. What is the z-score of x = –2, if it is 2.78 standard deviations to the right of the mean? 14. What is the z-score of x = 7, if it is 0.133 standard devi... |
mean. 29. In a normal distribution, x = –5 and z = –3.14. This tells you that x = –5 is ____ standard deviations to the ____ (right or left) of the mean. 30. In a normal distribution, x = 6 and z = –1.7. This tells you that x = 6 is ____ standard deviations to the ____ (right or left) of the mean. 31. About what perce... |
43. How would you represent the area to the left of one in a probability statement? Figure 6.12 44. What is the area to the right of one? Figure 6.13 45. Is P(x < 1) equal to P(x ≤ 1)? Why? 46. How would you represent the area to the left of three in a probability statement? Figure 6.14 47. What is the area to the rig... |
< x < __________) = __________ 59. Find the 70th percentile of the distribution for the time a CD player lasts. a. Sketch the situation. Label and scale the axes. Shade the region corresponding to the lower 70%. Figure 6.18 b. P(x < k) = __________ Therefore, k = _________ HOMEWORK 6.1 The Standard Normal Distribution... |
5 standard deviations below the mean, but that he believed his blood pressure was between 100 and 150 millimeters, what would you say to him? 65. Kyle’s doctor told him that the z-score for his systolic blood pressure is 1.75. Which of the following is the best interpretation of this standardized score? The systolic bl... |
700 and a second person took the ACT math test and scored 30, who did better with respect to the test they took? 6.2 Using the Normal Distribution Use the following information to answer the next two exercises: The patient recovery time from a particular surgical procedure is normally distributed with a mean of 5.3 da... |
IQ is normally distributed with a mean of 100 and a standard deviation of 15. Suppose one individual is randomly chosen. Let X = IQ of an individual. a. X ~ _____(_____,_____) b. Find the probability that the person has an IQ greater than 120. Include a sketch of the graph, and write a probability statement. c. MENSA ... |
X. probability statement. d. What percent of the children spend over ten hours per day unsupervised? e. Seventy percent of the children spend at least how long per day unsupervised? 78. In the 1992 presidential election, Alaska’s 40 election districts averaged 1,956.8 votes per district for President Clinton. The stan... |
. Table 6.3 displays the ordered real data (in minutes): 0.50 4.25 5 6 1.75 4.25 5.25 6 7.25 7.25 2 4.25 5.25 6.25 7.25 2.25 4.25 5.5 6.25 7.75 2.25 4.5 5.5 2.5 4.75 5.5 6.5 6.5 8 8.25 2.75 4.75 5.75 6.5 9.5 3.25 4.75 5.75 6.75 9.5 3.75 5 3.75 5 6 6 6.75 9.75 6.75 10.75 Table 6.3 a. Calculate the sample mean and the sa... |
ity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 6 | THE NORMAL DISTRIBUTION 363 40,000 40,000 45,050 45,500 46,249 48,134 49,133 50,071 50,096 50,466 50,832 51,100 51,500 51,900 52,000 52,132 52,200 52,530 52,692 53,864 54,000 55,000 55,000 55,000 55,000 55,000 55,000 55,082 57,000 58... |
The birth was uncomplicated, and the child needed no medical intervention. What is the probability that he was NOT the father? What is the probability that he could be the father? Calculate the z-scores first, and then use those to calculate the probability. 85. A NUMMI assembly line, which has been operating since 19... |
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