Section 7.3 Sampling Distributions and the Central Limit Theorem 311 Population standard deviation: �� = √ ��2 = √ 1.25 = 1.118 Now imagine tossing a tetrahedral die three times. The sequence of three numbers that is observed is a sample of size 3 drawn with replacement from the population just described. There are 64 possible samples, and they are all equally likely. Table 7.1 lists them and provides the value of the sample mean ̄x for each. Table 7.1 The 64 Possible Samples of Size 3 and Their Sample Means Sample ̄x Sample ̄x Sample ̄x Sample ̄x 1, 1, 1 1.00 2, 1, 1 1.33 3, 1, 1 1.67 4, 1, 1 2.00 1, 1, 2 1.33 2, 1, 2 1.67 3, 1, 2 2.00 4, 1, 2 2.33 1, 1, 3 1.67 2, 1, 3 2.00 3, 1, 3 2.33 4, 1, 3 2.67 1, 1, 4 2.00 2, 1, 4 2.33 3, 1, 4 2.67 4, 1, 4 3.00 1, 2, 1 1.33 2, 2, 1 1.67 3, 2, 1 2.00 4, 2, 1 2.33 1, 2, 2 1.67 2, 2, 2 2.00 3, 2, 2 2.33 4, 2, 2 2.67 1, 2, 3 2.00 2, 2, 3 2.33 3, 2, 3 2.67 4, 2, 3 3.00 1, 2, 4 2.33 2, 2, 4 2.67 3, 2, 4 3.00 4, 2, 4 3.33 1, 3, 1 1.67 2, 3, 1 2.00 3, 3, 1 2.33 4, 3, 1 2.67 1, 3, 2 2.00 2, 3, 2 2.33 3, 3, 2 2.67 4, 3, 2 3.00 1, 3, 3 2.33 2, 3, 3 2.67 3, 3, 3 3.00 4, 3, 3 3.33 1, 3, 4 2.67 2, 3, 4 3.00 3, 3, 4 3.33 4, 3, 4 3.67 1, 4, 1 2.00 2, 4, 1 2.33 3, 4, 1 2.67 4, 4, 1 3.00 1, 4, 2 2.33 2, 4, 2 2.67 3, 4, 2 3.00 4, 4, 2 3.33 1, 4, 3 2.67 2, 4, 3 3.00 3, 4, 3 3.33 4, 4, 3 3.67 1, 4, 4 3.00 2, 4, 4 3.33 3, 4, 4 3.67 4, 4, 4 4.00 The columns labeled ‘‘ ̄x ’’ contain the values of the sample mean for each of the 64 possible samples. Some of these values appear more than once, because several samples have the same mean. The mean of the sampling distribution is the average of these 64 values. The standard deviation of the sampling distribution is the population standard deviation of the 64 sample means, which can be computed by the method presented in Section 3.2. The mean and standard deviation are Mean: ��̄ x = 2.5 Standard deviation: �� = 0.6455 Comparing the mean and standard deviation of the sampling distribution to the population mean and standard deviation, we see that the mean ��̄ x of the sampling distribution is equal to the population mean ��. The standard deviation ��̄ x of the sampling distribution is 0.6455, which is less than the population standard deviation �� = 1.118. It is not immediately obvious how these two quantities are related. Note, however, that ��̄ x = 0.6455 = 1.118 √ 3 = �� √ 3 The sample size is n = 3, so ��̄ x = �� √ n . These relationships hold in general. Note that the standard deviation ��̄ x is sometimes called the standard error of the mean. SUMMARY Let ̄x be the mean of a simple random sample of size n, drawn from a population with mean �� and standard deviation ��. The mean of the sampling distribution is ��̄ x = ��. The standard deviation of the sampling distribution is ��̄ x = �� √ n . The standard deviation ��̄ x is sometimes called the standard error of the mean.
navidi_monk_elementary_statistics_2e_ch7-9
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