262 CHAPTER 5 Normal Probability Distributions Probability Distribution of Sample Means x f Probability 1 1 1 16 2 2 2 16 3 3 3 16 4 4 4 16 5 3 3 16 6 2 2 16 7 1 1 16 A Sampling Distribution of Sample Means The number of times four people go grocery shopping in a month is given by the population values 51, 3, 5, 76. A probability histogram for the data is shown at the left. You randomly choose two of the four people, with replacement. List all possible samples of size n = 2 and calculate the mean of each. These means form the sampling distribution of the sample means. Find the mean, variance, and standard deviation of the sample means. Compare your results with the mean m = 4, variance s 2 = 5, and standard deviation s = 25 ≈ 2.2 of the population. SOLUTION List all 16 samples of size 2 from the population and the mean of each sample. Sample Sample mean, x 1, 1 1 1, 3 2 1, 5 3 1, 7 4 3, 1 2 3, 3 3 3, 5 4 3, 7 5 Sample Sample mean, x 5, 1 3 5, 3 4 5, 5 5 5, 7 6 7, 1 4 7, 3 5 7, 5 6 7, 7 7 After constructing a probability distribution of the sample means, you can graph the sampling distribution using a probability histogram as shown at the left. Notice that the shape of the histogram is bell-shaped and symmetric, similar to a normal curve. The mean, variance, and standard deviation of the 16 sample means are mx = 4 Mean of the sample means 1sx2 2 = 5 2 = 2.5 Variance of the sample means and sx = A5 2 = 22.5 ≈ 1.6. Standard deviation of the sample means These results satisfy the properties of sampling distributions because mx = m = 4 and sx = s1 n = 2522 ≈ 1.6. TRY IT YOURSELF 1 List all possible samples of size n = 3, with replacement, from the population 51, 3, 56. Calculate the mean of each sample. Find the mean, variance, and standard deviation of the sample means. Compare your results with the mean m = 3, variance s 2 = 8 3, and standard deviation s = 28 3 ≈ 1.6 of the population. Answer: Page A39 EXAMPLE 1 To explore this topic further, see Activity 5.4 on page 274. 5.4 Study Tip Review Section 4.1 to find the mean and standard deviation of a probability distribution. Probability Number of grocery shopping trips 0.05 1 2 3 4 5 6 7 x 0.10 0.15 0.20 0.25 P(x) Probability Histogram of Population of x 1 2 3 4 5 6 7 Probability Sample mean 0.05 0.10 0.15 0.20 0.25 x P(x) Probability Histogram of Sampling Distribution of x
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