264 CHAPTER 5 Normal Probability Distributions Interpreting the Central Limit Theorem A study analyzed the sleep habits of college students. The study found that the mean sleep time was 6.9 hours, with a standard deviation of 1.5 hours. Random samples of 100 sleep times are drawn from this population, and the mean of each sample is determined. Find the mean and standard deviation of the sampling distribution of sample means. Then sketch a graph of the sampling distribution. (Adapted from National Institutes of Health) 6 8 10 11 3 2 4 7 9 5 x Individual sleep times (in hours) Distribution for All Sleep Times SOLUTION The mean of the sampling distribution is equal to the population mean, and the standard deviation of the sample means is equal to the population standard deviation divided by 2n. So, mx = m = 6.9 Mean of the sample means and sx = s2 n = 1.5 2 100 = 0.15. Standard deviation of the sample means Interpretation From the Central Limit Theorem, because the sample size is greater than 30, the sampling distribution can be approximated by a normal distribution with a mean of 6.9 hours and a standard deviation of 0.15 hour, as shown in the figure. x Mean of 100 sleep times (in hours) Distribution of Sample Means with n = 100 2 3 4 5 6 7 8 9 10 11 TRY IT YOURSELF 2 Random samples of size 64 are drawn from the population in Example 2. Find the mean and standard deviation of the sampling distribution of sample means. Then sketch a graph of the sampling distribution and compare it with the sampling distribution in Example 2. Answer: Page A39 EXAMPLE 2
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