SECTION 5.4 Sampling Distributions and the Central Limit Theorem 265 Interpreting the Central Limit Theorem Assume the training heart rates of all 20-year-old athletes are normally distributed, with a mean of 135 beats per minute and a standard deviation of 18 beats per minute, as shown in the figure. Random samples of size 4 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. 85 110 135 Rate (in beats per minute) 160 185 x Distribution of Population Training Heart Rates SOLUTION mx = m = 135 beats per minute Mean of the sample means and sx = s2 n = 182 4 = 9 beats per minute Standard deviation of the sample means Interpretation From the Central Limit Theorem, because the population is normally distributed, the sampling distribution of the sample means is also normally distributed, as shown in the figure. 85 110 135 Mean rate (in beats per minute) 160 185 x Distribution of Sample Means with n = 4 TRY IT YOURSELF 3 The diameters of fully grown white oak trees are normally distributed, with a mean of 3.5 feet and a standard deviation of 0.2 foot, as shown in the figure. Random samples of size 16 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. 3.5 Diameter (in feet) 3.7 3.9 4.1 2.9 3.1 3.3 x Distribution of Population Diameters Answer: Page A39 EXAMPLE 3 Picturing the World In a recent year, there were about 3.8 million parents in the United States who received child support payments.The histogram shows the distribution of children per custodial parent.The mean number of children was about 1.8 and the standard deviation was about 0.9. (Adapted from U.S. Census Bureau) 1 2 3 4 5 6 7 Probability Number of children 0.1 0.2 0.3 0.5 0.4 x Child Support P(x) You randomly select 35 parents who receive child support and ask how many children in their custody are receiving child support payments. What is the probability that the mean of the sample is between 1.5 and 1.9 children?
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