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STATS Modeling The World

David E. Bock, Paul F. Velleman, Richard D. De Veaux

Chapter 5

From the Data at Hand to the World at Large

Educators


Problem 1

Send money When they send out their fundraising letter, a philanthropic organization typically gets a return from about 5% of the people on their mailing list. To see what the response rate might be for future appeals, they did a simulation using samples of size 20, 50, 100, and 200. For each sample size, they simulated 1000 mailings with success rate p = 0.05 and constructed the histogram of the 1000 sample proportions. Explain how these four histograms demonstrate what the Central Limit Theorem says about the sampling distribution model for sample proportions. Be sure to talk about shape, center, and spread.

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Problem 2

Character recognition An automatic character recognition device can successfully read about 85% of handwritten credit card applications. To estimate what might happen when this device reads a stack of applications, the company did a simulation using samples of size 20, 50, 75, and 100. For each sample size, they simulated 1000 samples with success rate p = 0.85 and constructed the histogram of the 1000 sample proportions, shown here. Explain how these four histograms demonstrate what the Central Limit Theorem says about the sampling distribution model for sample proportions. Be sure to talk about shape, center, and spread.

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Problem 3

AP Exam A small class of five statistics students received the following scores on their AP Exam: 5, 4, 4, 3, 1.
a) Calculate the mean and standard deviation of these five scores.
b) List all possible sets of size 2 that could be chosen from this class. There are $_{5} \mathrm{C}_{2}=10$ such sets.
c) Calculate the mean of each of these sets of 2 scores and make a dotplot of the sampling distribution of the sample mean.
d) Calculate the mean and standard deviation of this sampling distribution. How do they compare to those of the individual scores? Is the sample mean an unbiased estimator of the population mean?

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Problem 4

AP Exam II For the small class described in Exercise 3 (mean $=3.4,$ sd $=1.517$ ), you will be selecting samples of size 3 (sampling without replacement).
a) List all possible samples of size 3 that could be chosen from this class.
b) Construct the sampling distribution of the sample mean for samples of size 3.
c) How do the mean and standard deviation of this sampling distribution compare to the mean and standard deviation of the population? To the mean and standard deviation of the sampling distribution of the sample mean from Exercise 3 part d?

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Problem 5

Marriage According to a Pew Research survey, about 27% of American adults are pessimistic about the future of marriage and the family. This is based on a sample, but assume that this percentage is correct for all American adults.
a) Using a binomial model, what is the probability that, in a sample of 20 American adults, 25% or fewer of the people in the sample are pessimistic about the future of marriage and family?
b) Now use a Normal model to compute that probability. How does this compare to your answer from part a?
c) Using a Binomial model, what is the probability that, in a sample of 700 American adults, 25% or fewer of the people of the people in the sample are pessimistic about the future of marriage and family?
d) Now use a Normal model to compute that probability. How does this compare to your answer from part c?
e) What do these answers tell you about the importance of checking that np and nq are both at least 10?

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Problem 6

Wow. Just, wow According to a 2013 poll from Public Policy Polling, 4% of American voters believe that shape-shifting reptilian people control our world by taking on human form and gaining power. Yes, you read that correctly! (This was a poll about conspiracy theories.) Assume that’s the actual proportion of Americans who hold that belief.
a) Use a binomial model to calculate the probability that, in a random sample of 100 people, at least 6% of those in the sample believe the thing about reptilian people controlling our world.
b) Use a Normal model to calculate the same probability. How does this compare with the answer in part a?
c) That same poll found that 51% of American voters believe there was a larger conspiracy responsible for the assassination of President Kennedy. Use a binomial model to calculate the probability that, in a random sample of 100 people, at least 57% of those in the
sample believe in the JFK conspiracy theory.
d) Use a normal model to calculate the same probability. How does this compare with the answer in part c?
e) What do these answers tell you about the importance of checking that np and nq are both at least 10?

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Problem 7

Send money, again The philanthropic organization in Exercise 1 expects about a 5% success rate when they send fundraising letters to the people on their mailing list. In Exercise 1 you looked at the histograms showing distributions of sample proportions from 1000 simulated mailings for samples of size 20, 50, 100, and 200. The sample statistics from each simulation were as follows:
$$\begin{array}{|l|l|} \hline{n} & \text { mean} & {\text { st. dev. }} \\ {20} & {0.0497} & {0.0479} \\ {50} & {0.0516} & {0.0309} \\ {100} & {0.0497} & {0.0215} \\ {200} & {0.0501} & {0.0152} \\ \hline\end{array}$$
a) According to the Central Limit Theorem, what should the theoretical mean and standard deviations be for these sample sizes?
b) How close are those theoretical values to what was observed in these simulations?
c) Looking at the histograms in Exercise 1, at what sample size would you be comfortable using the Normal model as an approximation for the sampling distribution?
d) What does the Success/Failure Condition say about the choice you made in part c?

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Problem 8

Character recognition, again The automatic character recognition device discussed in Exercise 2 successfully reads about 85% of handwritten credit card applications. In Exercise 2 you looked at the histograms showing distributions of sample proportions from 1000 simulated samples of size 20, 50, 75, and 100. The sample statistics from each simulation were as follows:
$$\begin{array}{|l|l|} \hline{n} & \text { mean} & {\text { st. dev. }} \\ {20} & {0.8481} & {0.0803} \\ {50} & {0.8507} & {0.0509} \\ {75} & {0.8481} & {0.0406} \\ {100} & {0.8488} & {0.0354} \\ \hline \end{array}$$
a) According to the Central Limit Theorem, what should the theoretical mean and standard deviations be for these sample sizes?
b) How close are those theoretical values to what was ob- served in these simulations?
c) Looking at the histograms in Exercise 2, at what sample size would you be comfortable using the Normal model as an approximation for the sampling distribution?
d) What does the Success/Failure Condition say about the choice you made in part c?

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Problem 9

Sample maximum The distribution of scores on a Statistics test for a particular class is skewed to the left. The professor wants to predict the maximum score and so wants to understand the distribution of the sample maximum. She simulates the distribution of the maxi- mum of the test for 30 different tests (with $n = 5$). The histogram below shows a simulated sampling distribution of the sample maximum from these tests.
a) Would a Normal model be a useful model for this sampling distribution? Explain.
b) The mean of this distribution is 46.3 and the SD is 3.5. Would you expect about 95% of the samples to have their maximums within 7 of 46.3? Why or why not?

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Problem 10

Soup A machine is supposed to fill cans with 16 oz of soup. Of course there, will be some variation in the
amount actually dispensed, and measurement errors are often approximately normally distributed. The manager would like to understand the variability of the variances of the samples, so he collects information from the last 250 batches of size 10 and plots a histogram of the variances:
a) Would a Normal model be a useful model for this sampling distribution? Explain.
b) The mean of this distribution is 0.009 and the SD is 0.004. Would you expect about 95% of the samples to have their variances within 0.008 of 0.009? Why or why not?

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Problem 11

Coin tosses In a large class of introductory Statistics students, the professor has each person toss a coin 16 times and calculate the proportion of his or her tosses that were heads. The students then report their results, and the professor plots a histogram of these several proportions.
a) What shape would you expect this histogram to be? Why?
b) Where do you expect the histogram to be centered?
c) How much variability would you expect among these proportions?
d) Explain why a Normal model should not be used here.

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Problem 12

M&M’s The candy company claims that 16% of the Milk Chocolate M&M’s it produces are green. Suppose that the candies are thoroughly mixed and then packaged in small bags containing about 50 M&M’s. A class of elementary school students learning about percents opens several bags, counts the various colors of the candies, and calculates the proportion that are green.
a) If we plot a histogram showing the proportions of green candies in the various bags, what shape would you expect it to have?
b) Can that histogram be approximated by a Normal model? Explain.
c) Where should the center of the histogram be?
d) What should the standard deviation of the sampling distribution be?

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Problem 13

More coins Suppose the class in Exercise 11 repeats the coin-tossing experiment.
a) The students toss the coins 25 times each. Use the 68–95– 99.7 Rule to describe the sampling distribution model.
b) Confirm that you can use a Normal model here.
c) They increase the number of tosses to 64 each. Draw and label the appropriate sampling distribution model. Check the appropriate conditions to justify your model.
d) Explain how the sampling distribution model changes as the number of tosses increases.

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Problem 14

Bigger bag Suppose the class in Exercise 12 buys bigger bags of candy, with 200 M&M’s each. Again the students calculate the proportion of green candies they find.
a) Explain why it’s appropriate to use a Normal model to describe the distribution of the proportion of green M&M’s they might expect.
b) Use the 68–95–99.7 Rule to describe how this proportion might vary from bag to bag.
c) How would this model change if the bags contained even more candies?

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Problem 15

Just (un)lucky? One of the students in the introductory Statistics class in Exercise 13 claims to have tossed her coin 200 times and found only 42% heads. What do you think of this claim? Explain.

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Problem 16

Too many green ones? In a really large bag of M&M’s, the students in Exercise 14 found 500 candies, and 18% of them were green. Is this an unusually large proportion of green M&M’s? Explain.

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Problem 17

Speeding State police believe that 70% of the drivers traveling on a major interstate highway exceed the speed limit. They plan to set up a radar trap and check the speeds of 80 cars.
a) Using the 68–95–99.7 Rule, draw and label the distribution of the proportion of these cars the police will observe speeding.
b) Do you think the appropriate conditions necessary for your analysis are met? Explain.

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Problem 18

Smoking Public health statistics for 2009 indicate that 20.6% of American adults smoke cigarettes. Using the 68–95–99.7 Rule, describe the sampling distribution model for the proportion of smokers among a randomly selected group of 50 adults. Be sure to discuss your assumptions and conditions.

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Problem 19

Vision It is generally believed that nearsightedness affects about 12% of all children. A school district has registered 170 incoming kindergarten children.
a) Can you apply the Central Limit Theorem to describe the sampling distribution model for the sample proportion of children who are nearsighted? Check the conditions and discuss any assumptions you need to make.
b) Sketch and clearly label the sampling model, based on the 68–95–99.7 Rule.
c) How many of the incoming students might the school expect to be nearsighted? Explain.

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Problem 20

Mortgages In July 2010, Lender Processing Services reported that homeowners were defaulting in record numbers; 12.4% of mortgages were delinquent or in foreclosure. Suppose a large bank holds 1731 adjustable-rate mortgages.
a) Can you apply the Central Limit Theorem to describe the sampling distribution model for the sample proportion of foreclosures? Check the conditions and discuss any assumptions you need to make.
b) Sketch and clearly label the sampling model, based on the 68–95–99.7 Rule.
c) How many of these homeowners might the bank expect will default on their mortgages? Explain.

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Problem 21

Loans Based on past experience, a bank believes that 7% of the people who receive loans will not make payments on time. The bank has recently approved 200 loans.
a) What are the mean and standard deviation of the proportion of clients in this group who may not make timely payments?
b) What assumptions underlie your model? Are the conditions met? Explain.
c) What’s the probability that over 10% of these clients will not make timely payments?

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Problem 22

Teens with phones Pew Research reported that, in 2013, 78% of all teens had a cell phone. Assume this estimate is correct.
a) We randomly pick 100 teens. Let $\hat{p}$ represent the proportion of teens in this sample who own a cell phone. What’s the appropriate model for the distribution of $\hat{p}$? Specify the name of the distribution, the mean, and the standard deviation. Be sure to verify that the conditions are met.
b) What’s the approximate probability that less than three fourths of this sample own a cell phone?

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Problem 23

Back to school? Best known for its testing program, ACT, Inc., also compiles data on a variety of issues in education. In 2004 the company reported that the national college freshman-to-sophomore retention rate held steady at 74% over the previous four years. Consider random samples of 400 freshmen who took the ACT. Use the 68–95–99.7 Rule to describe the sampling distribution model for the percentage of those students we expect to return to that school for their sophomore years. Do you think the appropriate conditions are met?

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Problem 24

Binge drinking A national study found that 44% of college students engage in binge drinking (5 drinks at a sitting for men, 4 for women). Use the 68–95–99.7 Rule to describe the sampling distribution model for the proportion of students in a randomly selected group of 200 college students who engage in binge drinking. Do you think the appropriate conditions are met?

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Problem 25

Back to school, again Based on the 74% national retention rate described in Exercise 23, does a college where 522 of the 603 freshman returned the next year as sophomores have a right to brag that it has an unusually high retention rate? Explain.

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Problem 26

Binge sample After hearing of Exercise 24’s national result that 44% of students engage in binge drinking (5 drinks at a sitting for men, 4 for women), a professor surveyed a random sample of 244 students and found that 96 of them admitted to binge drinking in the past week. Should he be surprised at this result? Explain.

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Problem 27

Polling Just before a referendum on a school budget, a local newspaper polls 400 voters in an attempt to predict whether the budget will pass. Suppose that the budget actually has the support of 52% of the voters. What’s the probability the newspaper’s sample will lead them to predict defeat? Be sure to verify that the assumptions and conditions necessary for your analysis are met.

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Problem 28

Seeds Information on a packet of seeds claims that the germination rate is 92%. What’s the probability that more than 95% of the 160 seeds in the packet will germinate? Be sure to discuss your assumptions and check the conditions that support your model.

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Problem 29

Gaydar Exercise 10 in Chapter 1 describes a study that showed that heterosexual women, during ovulation, were significantly better at correctly identifying the sexual orientation of a man from a photograph of his face than women who were not ovulating. In other words, ovulation improves a woman’s “gaydar.” Near ovulation, on average women correctly identified the orientation of about 65% of the 100 men shown to them. If this is the probability of correctly identifying the orientation of a man in any given photograph, what is the probability a woman would correctly classify 80 or more of the men (as two women in the study did)?

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Problem 30

Genetic defect It’s believed that 4% of children have a gene that may be linked to juvenile diabetes. Researchers hoping to track 20 of these children for several years test 732 newborns for the presence of this gene. What’s the probability that they find enough subjects for their study?

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Problem 31

“No Children” section Some restaurant owners, at the request of some of their less tolerant customers, have stopped allowing children into their restaurant. This, naturally, outrages other customers. One restaurateur hopes to please both sets of customers by having a “no children” section. She estimates that in her 120-seat restaurant, about 30% of her seats, on average, are taken by families with children. How many seats should be in the “children allowed” area in order to be very sure of having enough seating there? Comment on the assumptions and conditions that support your model, and explain what “very sure” means to you.

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Problem 32

Meals A restauranteur anticipates serving about 180 people on a Friday evening, and believes that about 20% of the patrons will order the chef’s steak special. How many of those meals should he plan on serving in order to be pretty sure of having enough steaks on hand to meet customer demand? Justify your answer, including an explanation of what “pretty sure” means to you.

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Problem 33

Sampling A sample is chosen randomly from a population that can be described by a Normal model.
a) What’s the sampling distribution model for the sample mean? Describe shape, center, and spread.
b) If we choose a larger sample, what’s the effect on this sampling distribution model?

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Problem 34

Sampling, part II A sample is chosen randomly from a population that was strongly skewed to the left.
a) Describe the sampling distribution model for the sample mean if the sample size is small.
b) If we make the sample larger, what happens to the sampling distribution model’s shape, center, and spread?
c) As we make the sample larger, what happens to the expected distribution of the data in the sample?

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Problem 35

Waist size A study measured the Waist Size of 250 men, finding a mean of 36.33 inches and a standard deviation of 4.02 inches. Here is a histogram of these measurements
a) Describe the histogram of Waist Size.
b) To explore how the mean might vary from sample to sample, they simulated by drawing many samples of size 2, 5, 10, and 20, with replacement, from the 250 measurements. Here are histograms of the sample means for each simulation. Explain how these histograms demonstrate what the Central Limit Theorem says about the sampling distribution model for sample means.

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Problem 36

CEO compensation In Chapter 5 we saw the distribution of the total compensation of the chief executive officers (CEOs) of the 800 largest U.S. companies (the Fortune 800). The average compensation (in thousands of dollars) is 10,307.31 and the standard deviation is 17,964.62. Here is a histogram of their annual compensations (in $1000):
a) Describe the histogram of Total Compensation. A research organization simulated sample means by drawing samples of 30, 50, 100, and 200, with replacement, from the 800 CEOs. The histograms show the distributions of means for many samples of each size.
b) Explain how these histograms demonstrate what the Central Limit Theorem says about the sampling distribution model for sample means. Be sure to talk about shape, center, and spread.
c) Comment on an oft-cited “rule of thumb” that “With a sample size of at least 30, the sampling distribution of the mean is Normal.”

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Problem 37

Waist size revisited Researchers measured the Waist Sizes of 250 men in a study on body fat. The true mean and standard deviation of the Waist Sizes for the 250 men are 36.33 in and 4.019 inches, respectively. In Exercise 35 you looked at the histograms of simulations that drew samples of sizes 2, 5, 10, and 20 (with replacement). The summary statistics for these simulations were as follows:
$$\begin{array}{|c|c|c|}\hline n & \text { mean } & {\text { st. dev. }} \\ \hline 2 & {36.314} & {2.855} \\ {5} & {36.314} & {1.805} \\ {10} & {36.341} & {1.276} \\ {20} & {36.339} & {0.895}\\ \hline \end{array}$$
a) According to the Central Limit Theorem, what should the theoretical mean and standard deviation be for each of these sample sizes?
b) How close are the theoretical values to what was observed in the simulation?
c) Looking at the histograms in Exercise 35, at what sample size would you be comfortable using the Normal model as an approximation for the sampling distribution?
d) What about the shape of the distribution of Waist Size explains your choice of sample size in part c?

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Problem 38

CEOs revisited In Exercise 36 you looked at the annual compensation for 800 CEOs, for which the true mean and standard deviation were (in thousands of dollars) 10,307.31 and 17,964.62, respectively. A simulation drew samples of sizes 30, 50, 100, and 200 (with replacement) from the total annual compensations of the Fortune 800 CEOs. The summary statistics for these simulations were as follows:
$$\begin{array}{|c|c|c|} \hline {n} & {\text { mean }} & {\text { st. dev. }} \\ \hline 30 & {10,251.73} & {3359.64} \\ {50} & {10,343.93} & {2483.84} \\ {100} & {10,329.94} & {1779.18} \\ {200} & {10,340.37} & {1230.79} \\ \hline\end{array}$$
a) According to the Central Limit Theorem, what should the theoretical mean and standard deviation be for each of these sample sizes?
b) How close are the theoretical values to what was observed from the simulation?
c) Looking at the histograms in Exercise 36, at what sample size would you be comfortable using the Normal model as an approximation for the sampling distribution?
d) What about the shape of the distribution of Total Compensation explains your answer in part c?

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Problem 39

GPAs A college’s data about the incoming freshmen indicates that the mean of their high school GPAs was 3.4, with a standard deviation of 0.35; the distribution was roughly mound-shaped and only slightly skewed. The students are randomly assigned to freshman writing seminars in groups of 25. What might the mean GPA of one of these seminar groups be? Describe the appropriate sampling distribution model—shape, center, and spread—with attention to assumptions and conditions. Make a sketch using the 68–95–99.7 Rule.

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Problem 40

Home values Assessment records indicate that the value of homes in a small city is skewed right, with a mean of $\$ 140,000$ and standard deviation of $\$ 60,000$. To check the accuracy of the assessment data, officials plan to conduct a detailed appraisal of 100 homes selected at random. Using the 68–95–99.7 Rule, draw and label an appropriate sampling model for the mean value of the homes selected.

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Problem 41

Lucky Spot? A reporter working on a story about the New York lottery contacted one of the authors of this book, wanting help analyzing data to see if some ticket sales outlets were more likely to produce winners. His data for each of the 966 New York lottery outlets are graphed below; the scatterplot shows the ratio TotalPaid/ TotalSales vs. TotalSales for the state’s “instant winner” games for all of 2007.
The reporter thinks that by identifying the outlets with the highest fraction of bets paid out, players might be able to increase their chances of winning. (Typically— but not always—instant winners are paid immediately (instantly) at the store at which they are purchased. However, the fact that tickets may be scratched off and then cashed in at any outlet may account for some outlets paying out more than they take in. The few with very low payouts may be on interstate highways where players may purchase cards but then leave.)
a) Explain why the plot has this funnel shape.
b) Explain why the reporter’s idea wouldn’t have worked anyway.

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Problem 42

Safe cities Allstate Insurance Company identified the 10 safest and 10 least-safe U.S. cities from among the 200 largest cities in the United States, based on the mean number of years drivers went between automobile accidents. The cities on both lists were all smaller than the 10 largest cities. Using facts about the sampling distribution model of the mean, explain why this is not surprising.

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Problem 43

Pregnancy Assume that the duration of human pregnancies can be described by a Normal model with mean 266 days and standard deviation 16 days.
a) What percentage of pregnancies should last between 270 and 280 days?
b) At least how many days should the longest 25% of all pregnancies last?
c) Suppose a certain obstetrician is currently providing prenatal care to 60 pregnant women. Let y represent the mean length of their pregnancies. According to the Central Limit Theorem, what’s the distribution of this sample mean, $\overline{y} ?$ Specify the model, mean, and standard deviation.
d) What’s the probability that the mean duration of these patients’ pregnancies will be less than 260 days?

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Problem 44

Rainfall Statistics from Cornell’s Northeast Regional Climate Center indicate that Ithaca, NY, gets an average of 35.4?? of rain each year, with a standard deviation of 4.2??. Assume that a Normal model applies.
a) During what percentage of years does Ithaca get more than 40?? of rain?
b) Less than how much rain falls in the driest 20% of all years?
c) A Cornell University student is in Ithaca for 4 years. Let $\overline{y}$ represent the mean amount of rain for those 4 years. Describe the sampling distribution model of this sample mean, $\overline{y}$.
d) What’s the probability that those 4 years average less than 30?? of rain?

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Problem 45

Pregnant again The duration of human pregnancies may not actually follow the Normal model described in Exercise 37.
a) Explain why it may be somewhat skewed to the left.
b) If the correct model is in fact skewed, does that change your answers to parts a, b, and c of Exercise 43? Explain why or why not for each.

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Problem 46

At work Some business analysts estimate that the length of time people work at a job has a mean of 6.2 years and a standard deviation of 4.5 years.
a) Explain why you suspect this distribution may be skewed to the right.
b) Explain why you could estimate the probability that 100 people selected at random had worked for their employers an average of 10 years or more, but you could not estimate the probability that an individual had done so.

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Problem 47

Dice and dollars You roll a die, winning nothing if the number of spots is odd, $\$ 1$ for a 2 or a 4, and $\$ 10$ for a 6.
a) Find the expected value and standard deviation of your prospective winnings.
b) You play twice. Find the mean and standard deviation of your total winnings.
c) You play 40 times. What’s the probability that you win at least $\$ 100$?

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Problem 48

New game You pay $\$ 10$ and roll a die. If you get a 6, you win $\$ 50$. If not, you get to roll again. If you get a 6 this time, you get your $\$ 10$ back.
a) Create a probability model for this game.
b) Find the expected value and standard deviation of your prospective winnings.
c) You play this game five times. Find the expected value and standard deviation of your average winnings.
d) 100 people play this game. What’s the probability the person running the game makes a profit?

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Problem 49

AP Stats 2012 The College Board reported the score distribution shown in the table for all students who took the 2012 AP Statistics exam.
$$\begin{array}{|l|l|}\hline{\text { Score}} & {\text {Percent of Students }} \\ \hline {5} & {12.5} \\ {4} & {21.1} \\ {3} & {25.6} \\ {2} & {18.0} \\ {1} & {22.8} \\ \hline\end{array}$$
a) Find the mean and standard deviation of the scores.
b) If we select a random sample of 40 AP Statistics students, would you expect their scores to follow a Normal model? Explain.
c) Consider the mean scores of random samples of 40 AP Statistics students. Describe the sampling model for these means (shape, center, and spread).

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Problem 50

Museum membership A museum offers several levels of membership, as shown in the table.
$$\begin{array}{|l|l|l|} \hline {\text { Member}} & \text { Amount} & {\text {of Percent of }} \\ {\text { Category}} &\text { Donation (s)} & {\text {Members }} \\ {\text { Individual }} & {50} & {41} \\ {\text { Family }} & {100} & {37} \\ {\text { Sponsor }} & {250} & {14} \\ {\text { Patron }} & {500} & {7} \\ {\text { Benefactor }} & {1000} & {1}\\ \hline \end{array}$$
a) Find the mean and standard deviation of the donations.
b) During their annual membership drive, they hope to sign up 50 new members each day. Would you expect the distribution of the donations for a day to follow a Normal model? Explain.
c) Consider the mean donation of the 50 new members each day. Describe the sampling model for these means (shape, center, and spread).

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Problem 51

AP Stats 2012, again An AP Statistics teacher had 63 students preparing to take the AP exam discussed in Exercise 49. Though they were obviously not a random sample, he considered his students to be “typical” of all the national students. What’s the probability that his students will achieve an average score of at least 3?

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Problem 52

Joining the museum One of the museum’s phone volunteers sets a personal goal of getting an average donation of at least $100 from the new members she enrolls during the membership drive described in Exercise 50. If she gets 80 new members and they can be considered a random sample of all the museum’s members, what is the probability that she can achieve her goal?

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Problem 53

Pollution Carbon monoxide (CO) emissions for a certain kind of car vary with mean 2.9 g>mi and standard deviation 0.4 $\mathrm{g} / \mathrm{mi}$. A company has 80 of these cars in its fleet.Let $\overline{y}$ represent the mean CO level for the company’s fleet.
a) What’s the approximate model for the distribution of $\overline{y} ?$ Explain.
b) Estimate the probability that $\overline{y}$ is between 3.0 and 3.1 $\mathrm{g} / \mathrm{mi}$.
c) There is only a 5% chance that the fleet’s mean CO level is greater than what value?

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Problem 54

Potato chips The weight of potato chips in a mediumsize bag is stated to be 10 ounces. The amount that the packaging machine puts in these bags is believed to have a Normal model with mean 10.2 ounces and standard deviation 0.12 ounces.
a) What fraction of all bags sold are underweight?
b) Some of the chips are sold in “bargain packs” of 3 bags. What’s the probability that none of the 3 is underweight?
c) What’s the probability that the mean weight of the 3 bags is below the stated amount?
d) What’s the probability that the mean weight of a 24-bag case of potato chips is below 10 ounces?

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Problem 55

Tips A waiter believes the distribution of his tips has a model that is slightly skewed to the right, with a mean of $\$ 9.60$ and a standard deviation of $\$5.40$.
a) Explain why you cannot determine the probability that a given party will tip him at least $\$20$.
b) Can you estimate the probability that the next 4 parties will tip an average of at least $\$15$? Explain.
c) Is it likely that his 10 parties today will tip an average of at least $\$15$? Explain.

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Problem 56

Groceries A grocery store’s receipts show that Sunday customer purchases have a skewed distribution with a mean of $\$32$ and a standard deviation of $\$20$.
a) Explain why you cannot determine the probability that the next Sunday customer will spend at least $\$40$.
b) Can you estimate the probability that the next 10 Sunday customers will spend an average of at least$\$40$? Explain.
c) Is it likely that the next 50 Sunday customers will spend an average of at least $\$40$? Explain.

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Problem 57

More tips The waiter in Exercise 55 usually waits on about 40 parties over a weekend of work.
a) Estimate the probability that he will earn at least $\$500$ in tips.
b) How much does he earn on the best 10% of such weekends?

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Problem 58

More groceries Suppose the store in Exercise 56 had 312 customers this Sunday.
a) Estimate the probability that the store’s revenues were at least $\$10,000$.
b) If, on a typical Sunday, the store serves 312 customers, how much does the store take in on the worst 10% of such days?

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Problem 59

IQs Suppose that IQs of East State University’s students can be described by a Normal model with mean 130 and standard deviation 8 points. Also suppose that IQs of students from West State University can be described by a Normal model with mean 120 and standard deviation 10.
a) We select a student at random from East State. Find the probability that this student’s IQ is at least 125 points.
b) We select a student at random from each school. Find the probability that the East State student’s IQ is at least 5 points higher than the West State student’s IQ.
c) We select 3 West State students at random. Find the probability that this group’s average IQ is at least 125 points.
d) We also select 3 East State students at random. What’s the probability that their average IQ is at least 5 points higher than the average for the 3 West Staters?

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Problem 60

Milk Although most of us buy milk by the quart or gallon, farmers measure daily production in pounds. Ayrshire cows average 47 pounds of milk a day, with a standard deviation of 6 pounds. For Jersey cows, the mean daily production is 43 pounds, with a standard deviation of 5 pounds. Assume that Normal models describe milk production for these breeds.
a) We select an Ayrshire at random. What’s the probability that she averages more than 50 pounds of milk a day?
b) What’s the probability that a randomly selected Ayrshire gives more milk than a randomly selected Jersey?
c) A farmer has 20 Jerseys. What’s the probability that the average production for this small herd exceeds 45 pounds of milk a day?
d) A neighboring farmer has 10 Ayrshires. What’s the probability that his herd average is at least 5 pounds higher than the average for part c’s Jersey herd?

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