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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? c) What do these answers tell you about the importance of checking that \(n p\) and \(n q\) are both at least \(10 ?\)

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 \(50 \mathrm{M} \& \mathrm{M}^{\prime} \mathrm{s}\). 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?

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.

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?

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?

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?

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"

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 \(\bar{y}\) represent the mean length of their pregnancies. According to the Central Limit Theorem, what's the distribution of this sample mean, \(\bar{y}\) ? 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?

The weight of potato chips in a medium size 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?

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?