312 Chapter 7 The Normal Distribution EXAMPLE 7.19 Find the mean and standard deviation of a sampling distribution Among students at a certain college, the mean number of hours of television watched per week is �� = 10.5, and the standard deviation is �� = 3.6. A simple random sample of 16 students is chosen for a study of viewing habits. Let ̄x be the mean number of hours of TV watched by the sampled students. Find the mean ��̄ x and the standard deviation ��̄ x of ̄x. Solution The mean of ̄x is ��̄ x = �� = 10.5 The sample size is n = 16. Therefore, the standard deviation of ̄x is ��̄ x = �� √ n = 3.6 √ 16 = 0.9 It makes sense that the standard deviation of ̄x is less than the population standard deviation ��. In a sample, it is unusual to get all large values or all small values. Samples usually contain both large and small values that cancel each other out when the sample mean is computed. For this reason, the distribution of ̄x is less spread out than the population distribution. Therefore, the standard deviation of ̄x is less than the population standard deviation. Check Your Understanding 1. A population has mean �� = 6 and standard deviation �� = 4. Find ��̄ x and ��̄ x for samples of size n = 25. ��̄ x = 6; ��̄ x = 0.8 2. A population has mean �� = 17 and standard deviation �� = 20. Find ��̄ x and ��̄ x for samples of size n = 100. ��̄ x = 17; ��̄ x = 2.0 Answers are on page 318. The probability histogram for the sampling distribution of x Consider again the example of the tetrahedral die. Let us compare the probability distribution for the population and the sampling distribution. The population consists of the numbers 1, 2, 3, and 4, each of which is equally likely. The sampling distribution for ̄x can be determined from Table 7.1. The probability that the sample mean is 1.00 is 1∕64, because out of the 64 possible samples, only one has a sample mean equal to 1.00. Similarly, the probability that ̄x = 1.33 is 3∕64, because there are three samples out of 64 whose sample mean is 1.33. Figure 7.20 presents the probability histogram of the population and Figure 7.21 presents the sampling distribution for ̄x. 1 2 3 4 0.30 0.25 0.20 0.15 0.10 0.05 0 Probability Figure 7.20 Probability histogram for the population 1 2 3 4 0.20 0.15 0.10 0.05 0 Probability Figure 7.21 Probability histogram for the sampling distribution of ̄x for samples of size 3 Note that the probability histogram for the sampling distribution looks a lot like the normal curve, whereas the probability histogram for the population does not. Remarkably, it is true that, for any population, if the sample size is large enough, the sample mean ̄x will be approximately normally distributed. For a symmetric population like the one
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