6-3 Sampling Distributions and Estimators 281 2. Sampling with Replacement The Pew Research Center conducts many different surveys in the United States each year. a. Do you think that for each individual survey, the respondents are randomly selected with replacement or without replacement? b. Give two reasons why statistical methods tend to be based on the assumption that sampling is conducted with replacement, instead of without replacement. 3. Unbiased Estimators Data Set 1 “Body Data” in Appendix B includes systolic blood pressure measurements from 147 adult females. If we compute the values of sample statistics from that sample, which of the following statistics are unbiased estimators of the corresponding population parameters: sample mean; sample median; sample range; sample variance; sample standard deviation; sample proportion? 4. Sampling Distribution Data Set 1 “Body Data” in Appendix B includes systolic blood pressure measurements from 147 adult females. If we explore this sample by constructing a histogram and finding the mean and standard deviation, do those results describe the sampling distribution of the mean? Why or why not? 5. Good Sample? An economist is investigating the incomes of college students. Because she lives in Maine, she obtains sample data from that state. Is the resulting mean income of college students a good estimator of the mean income of college students in the United States? Why or why not? 6. College Presidents There are about 4200 college presidents in the United States, and they have annual incomes with a distribution that is skewed instead of being normal. Many different samples of 40 college presidents are randomly selected, and the mean annual income is computed for each sample. a. What is the approximate shape of the distribution of the sample means (uniform, normal, skewed, other)? b. What value do the sample means target? That is, what is the mean of all such sample means? In Exercises 7–10, use the same population of {4, 5, 9} that was used in Examples 2 and 5. As in Examples 2 and 5, assume that samples of size n = 2 are randomly selected with replacement. 7. Sampling Distribution of the Sample Variance a. Find the value of the population variance s 2. b. Table 6-2 describes the sampling distribution of the sample mean. Construct a similar table representing the sampling distribution of the sample variance s 2. Then combine values of s 2 that are the same, as in Table 6-3 (Hint: See Example 2 on page 276 for Tables 6-2 and 6-3, which describe the sampling distribution of the sample mean.) c. Find the mean of the sampling distribution of the sample variance. d. Based on the preceding results, is the sample variance an unbiased estimator of the population variance? Why or why not? 8. Sampling Distribution of the Sample Standard Deviation For the following, round results to three decimal places. a. Find the value of the population standard deviation s. b. Table 6-2 describes the sampling distribution of the sample mean. Construct a similar table representing the sampling distribution of the sample standard deviation s. Then combine values of s that are the same, as in Table 6-3 (Hint: See Example 2 on page 276 for Tables 6-2 and 6-3, which describe the sampling distribution of the sample mean.) c. Find the mean of the sampling distribution of the sample standard deviation. d. Based on the preceding results, is the sample standard deviation an unbiased estimator of the population standard deviation? Why or why not?
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