6-4 The Central Limit Theorem 283 question having 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? 18. Hybridization A hybridization experiment begins with four peas having yellow pods and one pea having a green pod. Two of the peas are randomly selected with replacement from this population. a. After identifying the 25 different possible samples, find the proportion of peas with yellow pods in each of them, then construct a table to describe the sampling distribution of the proportions of peas with yellow pods. 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 peas with yellow pods? Does the mean of the sampling distribution of proportions always equal the population proportion? 19.Using a Formula to Describe a Sampling Distribution Exercise 15 “Births” 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 (represented by x) of 0, 0.5, and 1. Based on the results, does the formula describe the sampling distribution? Why or why not? P1x2 = 1 212 - 2x2!12x2! where x = 0, 0.5, 1 20.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.) 6-3 Beyond the Basics Key Concept In the preceding section we saw that the sampling distribution of sample means tends to be a normal distribution as the sample size increases. In this section we introduce and apply the central limit theorem. The central limit theorem allows us to use a normal distribution for some very meaningful and important applications. Given any population with any distribution (uniform, skewed, whatever), the distribution of sample means x can be approximated by a normal distribution when the samples are large enough with n 7 30. (There are some special cases of very nonnormal distributions for which the requirement of n 7 30 isn’t quite enough, so the number 30 should be higher in those cases, but those cases are rare.) 6-4 The Central Limit Theorem
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