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69 . The mean of a normally-distributed population is 50, and the standard deviation is four. If you draw 100 samples of size 40 from this population, describe what you would expect to see in terms of the sampling distribution of the sample mean.
70 . X is a random variable with a mean of 25 and a standard deviation of two. Write the distribution for the sample mean of samples of size 100 drawn from this population.
71 . Your friend is doing an experiment drawing samples of size 50 from a population with a mean of 117 and a standard deviation of 16. This sample size is large enough to allow use of the central limit theorem, so he says the standard deviation of the sampling distribution of sample means will also be 16. Explain why this is wrong, and calculate the correct value.
72 . You are reading a research article that refers to “the standard error of the mean.” What does this mean, and how is it calculated?
Use the following information to answer the next six exercises. You repeatedly draw samples of n = 100 from a population with a mean of 75 and a standard deviation of 4.5.
73 . What is the expected distribution of the sample means?
74 . One of your friends tries to convince you that the standard error of the mean should be 4.5. Explain what error your friend made.
75 . What is the z -score for a sample mean of 76?
76 . What is the z -score for a sample mean of 74.7?
77 . What sample mean corresponds to a z -score of 1.5?
78 . If you decrease the sample size to 50, will the standard error of the mean be smaller or larger? What would be its value?
Use the following information to answer the next two questions. We use the empirical rule to analyze data for samples of size 60 drawn from a population with a mean of 70 and a standard deviation of 9.
79 . What range of values would you expect to include 68 percent of the sample means?
80 . If you increased the sample size to 100, what range would you expect to contain 68 percent of the sample means, applying the empirical rule?
81 . How does the central limit theorem apply to sums of random variables?
82 . Explain how the rules applying the central limit theorem to sample means, and to sums of a random variable, are similar.
83 . If you repeatedly draw samples of size 50 from a population with a mean of 80 and a standard deviation of four, and calculate the sum of each sample, what is the expected distribution of these sums?
Use the following information to answer the next four exercises. You draw one sample of size 40 from a population with a mean of 125 and a standard deviation of seven.
84 . Compute the sum. What is the probability that the sum for your sample will be less than 5,000?
85 . If you drew samples of this size repeatedly, computing the sum each time, what range of values would you expect to contain 95 percent of the sample sums?
86 . What value is one standard deviation below the mean?
87 . What value corresponds to a z -score of 2.2?
88 . What does the law of large numbers say about the relationship between the sample mean and the population mean?
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