Section 8.1 Confidence Intervals for a Population Mean, Standard Deviation Known 349 as 67.30 ± 1. If we think that ̄x = 67.30 could be off by as much as 10 points from the population mean, we would estimate �� with the interval 57.30 < �� < 77.30, which could also be written as 67.30 ± 10. The plus-or-minus number is called the margin of error. We need to determine how large to make the margin of error so that the interval is likely to contain the population mean. To do this, we use the sampling distribution of ̄x. Explain It Again The symbol ±: The symbol ± means to form an interval by adding and subtracting. For example, 67.30 ± 1 means the interval from 67.30 − 1 to 67.30 + 1, or, in other words, from 66.30 to 68.30. Because the sample size is large (n > 30), the Central Limit Theorem tells us that the sampling distribution of ̄x is approximately normal with mean �� and standard deviation (also called the standard error) given by Standard error = �� √ n = 15 √ 100 = 1.5 √ n is Recall: The quantity ��∕ called the standard error of the mean. We will now construct a 95% confidence interval for ��. We begin with a normal curve, and using the method described in Section 7.1 (Example 7.11), we find the z-scores that bound the middle 95% of the area under the curve. These z-scores are 1.96 and −1.96 (see Figure 8.1). The value 1.96 is called the critical value. To obtain the margin of error, we multiply the critical value by the standard error. Margin of error = Critical value ⋅ Standard error = (1.96)(1.5) = 2.94 Area = 0.025 Area = 0.025 Area = 0.95 −1.96 1.96 Figure 8.1 95% of the area under the standard normal curve is between z = −1.96 and z = 1.96. A 95% confidence interval for �� is therefore ̄x − 2.94 < �� < ̄x + 2.94 67.30 − 2.94 < �� < 67.30 + 2.94 64.36 < �� < 70.24 There are several ways to express this confidence interval.Wecan write 64.36 < �� < 70.24, 67.30 ± 2.94, or (64.36, 70.24). In words, we would say, ‘‘We are 95% confident that the population mean is between 64.36 and 70.24.’’ Figures 8.2 and 8.3 help explain why this interval is called a 95% confidence interval. Figure 8.2 illustrates a sample whose mean ̄x is in the middle 95% of its distribution. The 95% confidence interval constructed from this value of ̄x covers the true population mean ��. 95% μ x μ − 2.94 μ + 2.94 x − 2.94 x + 2.94 Figure 8.2 The sample mean ̄x comes from the middle 95% of the distribution, so the 95% confidence interval ̄x ± 2.94 succeeds in covering the population mean ��. Figure 8.3 illustrates a sample whose mean ̄x is in one of the tails of the distribution, outside the middle 95%. The 95% confidence interval constructed from this value of ̄x does not cover the true population mean ��. 95% x μ − 2.94 μ μ + 2.94 x − 2.94 x + 2.94 Figure 8.3 The sample mean ̄x comes from the outer 5% of the distribution, so the 95% confidence interval ̄x ± 2.94 fails to cover the population mean ��.
navidi_monk_elementary_statistics_2e_ch7-9
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