Solution: In Exercises 1114, use the population of ages {56, 49, 58, 46} of the four

Minting Quarters In a recent year, the U.S. Mint in Denver manufactured 270 million quarters. Assume that on each day of production, a sample of 50 quarters is randomly selected, and the mean weight is obtained. a. Given that the population of quarters has a mean weight of 5.67 g, what do you know about the mean of the sample means? b. What do you know about the shape of the distribution of the sample means? c. The population of quarters has a mean of 5.67 g, but the weights of individual quarters vary. For each sample of 50 quarters, consider the proportion of quarters that weigh less than 5.67 g. What do you know about the shape of the distribution of the sample proportions?

. Sampling with Replacement In a recent year, the U.S. Mint in Denver manufactured 270 million quarters. As part of the mint's quality control program, samples of quarters are randomly selected each day for detailed inspection to confirm that they meet all required specifications. a. Do you think the quarters 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.

Unbiased Estimators Data Set 1 in Appendix B includes a sample of 40 pulse rates of women. 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?

Sampling Distribution Data Set 20 in Appendix B includes a sample of weights of 100 M&M candies. If we explore this sample of 100 weights 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?

. Good Sample? For the population of all college students currently taking a statistics course, you want to estimate the proportion who are women. You obtain a simple random sample of statistics students at Broward College in Florida. Is the resulting sample proportion a good estimator of the population proportion of all college statistics students? Why or why not?

Lottery Results Many states have a Pick 3 lottery in which three digits are randomly selected each day. Winning requires that you select the same three digits in the same order that they are drawn. Assume that you compute the mean of each set of three selected digits. 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? c. For each set of three digits that is selected, if you find the proportion of odd numbers, what is the mean of those proportions?

In Exercises 710, use the same population of {4, 5, 9} that was used in Examples 1 and 5. As in Examples 1 and 5, assume that samples of size n = 2 are randomly selected with replacement.Sampling Distribution of the Sample Variance a. Find the value of the population variance 2 .b. Table 63 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 64. (Hint: See Example 1 for Tables 63 and 64 that 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?

In Exercises 710, use the same population of {4, 5, 9} that was used in Examples 1 and 5. As in Examples 1 and 5, assume that samples of size n = 2 are randomly selected with replacement.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 . b. Table 63 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 64. (Hint: See Example 1 for Tables 63 and 64 that 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?

In Exercises 710, use the same population of {4, 5, 9} that was used in Examples 1 and 5. As in Examples 1 and 5, assume that samples of size n = 2 are randomly selected with replacement.. Sampling Distribution of the Sample Median a. Find the value of the population median. b. Table 63 describes the sampling distribution of the sample mean. Construct a similar table representing the sampling distribution of the sample median. Then combine values of the median that are the same, as in Table 64. (Hint: See Example 1 for Tables 63 and 64 that describe the sampling distribution of the sample mean.) c. Find the mean of the sampling distribution of the sample median. d. Based on the preceding results, is the sample median an unbiased estimator of the population median? Why or why not?

In Exercises 710, use the same population of {4, 5, 9} that was used in Examples 1 and 5. As in Examples 1 and 5, assume that samples of size n = 2 are randomly selected with replacement.Sampling Distribution of the Sample Proportion a. For the population, find the proportion of odd numbers. b. Table 63 describes the sampling distribution of the sample mean. Construct a similar table representing the sampling distribution of the sample proportion of odd numbers. Then combine values of the sample proportion that are the same, as in Table 64. (Hint: See Example 1 for Tables 63 and 64 that describe the sampling distribution of the sample mean.) c. Find the mean of the sampling distribution of the sample proportion of odd numbers. d. Based on the preceding results, is the sample proportion an unbiased estimator of the population proportion? Why or why not?

In Exercises 1114, use the population of ages {56, 49, 58, 46} of the four U.S. presidents (Lincoln, Garfield, McKinley, Kennedy) when they were assassinated in office. Assume that random samples of size n = 2 are selected with replacement.. Sampling Distribution of the Sample Mean a. After identifying the 16 different possible samples, find the mean of each sample, then construct a table representing the sampling distribution of the sample mean. In the table, combine values of the sample mean that are the same. (Hint: See Table 64 in Example 1.) b. Compare the mean of the population {56, 49, 58, 46} to the mean of the sampling distribution of the sample mean. c. Do the sample means target the value of the population mean? In general, do sample means make good estimators of population means? Why or why not?

In Exercises 1114, use the population of ages {56, 49, 58, 46} of the four U.S. presidents (Lincoln, Garfield, McKinley, Kennedy) when they were assassinated in office. Assume that random samples of size n = 2 are selected with replacement.Sampling Distribution of the Median Repeat Exercise 11using medians instead of means.

In Exercises 1114, use the population of ages {56, 49, 58, 46} of the four U.S. presidents (Lincoln, Garfield, McKinley, Kennedy) when they were assassinated in office. Assume that random samples of size n = 2 are selected with replacement.Sampling Distribution of the Range Repeat Exercise 11 using ranges instead of means.

In Exercises 1114, use the population of ages {56, 49, 58, 46} of the four U.S. presidents (Lincoln, Garfield, McKinley, Kennedy) when they were assassinated in office. Assume that random samples of size n = 2 are selected with replacement. Sampling Distribution of the Variance Repeat Exercise 11 using variances instead of means.

Births: Sampling Distribution of Sample Proportion When two births are randomly selected, the sample space for genders is bb, bg, gb, and gg. Assume that those four outcomes are equally likely. Construct a table that describes the sampling distribution of the sample proportion of girls from two births. Does the mean of the sample proportions equal the proportion of girls in two births? Does the result suggest that a sample proportion is an unbiased estimator of a population proportion?

Births: Sampling Distribution of Sample Proportion When three births are randomly selected, the sample space for genders is bbb, bbg, bgb, bgg, gbb, gbg, ggb, and ggg. Assume that those eight outcomes are equally likely. Construct a table that describes the sampling distribution of the sample proportion of girls from three births. Does the mean of the sample proportions equal the proportion of girls in three births?

. SAT and ACT Tests Because they enable efficient procedures for evaluating answers, multiplechoice questions are commonly used on standardized tests, such as the SAT or ACT. Such questions typically have five choices, one of which is correct. Assume that you must make random guesses for two such questions. Assume that both questions have correct answers of a. a. After listing the 25 different possible samples, find the proportion of correct answers in each sample, then construct a table that describes the sampling distribution of the sample proportions of correct responses. b. Find the mean of the sampling distribution of the sample proportion. c. Is the mean of the sampling distribution (from part (b)) equal to the population proportion of correct responses? Does the mean of the sampling distribution of proportions always equal the population proportion?

. Quality Control After constructing a new manufacturing machine, five prototype integrated circuit chips are produced and it is found that two are defective (D) and three are acceptable (A). Assume that two of the chips are randomly selected with replacement from this population. a. After identifying the 25 different possible samples, find the proportion of defects in each of them, then use a table to describe the sampling distribution of the proportions of defects. b. Find the mean of the sampling distribution. c. Is the mean of the sampling distribution (from part (b)) equal to the population proportion of defects? Does the mean of the sampling distribution of proportions always equal the population proportion?

Using a Formula to Describe a Sampling Distribution Exercise 15 requires the construction of a table that describes the sampling distribution of the proportions of girls from two births. Consider the formula shown here, and evaluate that formula using sample proportions x of 0, 0.5, and 1. Based on the results, does the formula describe the sampling distribution? Why or why not? P ( x ) = 1 2 ( 2 2 x ) ! ( 2 x ) ! where x = 0 , 0.5 , 1

Mean Absolute Deviation Is the mean absolute deviation of a sample a good statistic for estimating the mean absolute deviation of the population? Why or why not? (Hint: See Example 5.)