350 Chapter 8 Confidence Intervals Now in the long run, 95% of the sample means we observe will be like the one in Figure 8.2. They will be in the middle 95% of the distribution and their confidence intervals will cover the population mean. Therefore, in the long run, 95% of the confidence intervals we construct will cover the population mean. Only 5% of the sample means we observe will be outside the middle 95% of the distribution like the one in Figure 8.3. So in the long run, only 5% of the confidence intervals we construct will fail to cover the population mean. To summarize, the confidence interval we just constructed is a 95% confidence interval, because the method we used to construct it will cover the population mean �� for 95% of all the possible samples that might be drawn. We can also say that the interval has a confidence level of 95%. DEFINITION ∙ A confidence interval is an interval that is used to estimate the value of a parameter. ∙ The confidence level is a percentage between 0% and 100% that measures the success rate of the method used to construct the confidence interval. If we were to draw many samples and use each one to construct a confidence interval, then in the long run, the percentage of confidence intervals that cover the true value would be equal to the confidence level. EXAMPLE 8.1 Construct and interpret a 95% confidence interval A large sample has mean ̄x = 7.1 and standard error ��∕ √ n = 2.3. Find the margin of error for a 95% confidence interval. Construct a 95% confidence interval for the population mean �� and explain what it means to say that the confidence level is 95%. Solution As shown in Figure 8.1, the critical value for a 95% confidence interval is 1.96. Therefore, the margin of error is Margin of error = Critical value ⋅ Standard error = (1.96)(2.3) = 4.5 The point estimate of �� is ̄x = 7.1. To construct a confidence interval, we add and subtract the margin of error from the point estimate. So the 95% confidence interval is 7.1 ± 4.5. We can also write this as 7.1 − 4.5 < �� < 7.1 + 4.5, or 2.6 < �� < 11.6. The level of this confidence interval is 95% because if we were to draw many samples and use this method to construct the corresponding confidence intervals, then in the long run, 95% of the intervals would cover the true value of the population mean ��. Unless we were unlucky in the sample we drew, the population mean �� will be between 2.6 and 11.6. Objective 2 Find critical values for confidence intervals Finding the critical value for a given confidence level Although 95% is the most commonly used confidence level, sometimes we will want to construct a confidence interval with a different level. We can construct a confidence interval with any confidence level between 0% and 100% by finding the appropriate critical value for that level. We have seen that the critical value for a 95% confidence interval is z = 1.96, because 95% of the area under a normal curve is between z = −1.96 and z = 1.96. Similarly, the critical value for a 99% confidence interval is the z-score for which the area between z and −z is 0.99, the critical value for a 98% confidence interval is the z-score for which the area between z and −z is 0.98, and so on. The row of Table A.3 labeled ‘‘z’’ presents critical values for several confidence levels. Following is part of that row, which presents critical values for four of the most commonly used confidence levels. z · · · 1.645 1.96 2.326 2.576 · · · · · · 90% 95% 98% 99% ··· Confidence level
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