356 Chapter 8 Confidence Intervals Solution We check the assumptions. The sample is a simple random sample, and the population is known to be normal. The assumptions are met, so we may proceed. Step 1: Find the point estimate. The point estimate is the sample mean ̄x = 20.25. Step 2: Find the critical value z��∕2. The desired confidence level is 90%. We look on the last line of Table A.3 and find that the critical value is z��∕2 = 1.645. Step 3: Find the standard error and the margin of error. The standard error is �� √ n = 0.2 √ 6 = 0.08165 We multiply the standard error by the critical value to obtain the margin of error: Margin of error = z��∕2 �� √ n = (1.645)(0.08165) = 0.1343 Step 4: Construct the confidence interval. The 90% confidence interval is ̄x − z��∕2 �� √ n < �� < ̄x + z��∕2 �� √ n 20.25 − 0.1343 < �� < 20.25 + 0.1343 20.12 < �� < 20.38 (rounded to two decimal places, like ̄x) Step 5: Interpret the result. We are 90% confident that the mean weight �� is between 20.12 and 20.38. Another way to say this is that we are 90% confident that the mean weight �� is in the interval 20.25 ± 0.13. If we were to draw many different samples and use this method to construct the corresponding confidence intervals, then in the long run, 90% of them would cover the true population mean ��. So unless we were somewhat unlucky in the sample we drew, the true mean weight is between 20.12 and 20.38 ounces. Explain It Again The relationship between confidence and the margin of error: If we want to increase our confidence that an interval contains the true value, we must increase the critical value. This increases the margin of error, which makes the confidence interval wider. A confidence level of 90% is the lowest level commonly used in practice. In Example 8.6, we will construct a 99% confidence interval. EXAMPLE 8.6 Construct a confidence interval Use the data in Example 8.5 to construct a 99% confidence interval for the mean fill weight. Compare the margin of error of this confidence interval to the 90% confidence interval constructed in Example 8.5. Solution As in Example 8.5, the assumptions are met, so we may proceed. Step 1: Find the point estimate. The point estimate is ̄x = 20.25. Step 2: Find the critical value z��∕2. The desired level is 99%. We look on the last line of Table A.3 and find that z��∕2 = 2.576. Step 3: Find the standard error and the margin of error. The standard error is �� √ n = 0.2 √ 6 = 0.08165 We multiply the standard error by the critical value to obtain the margin of error: Margin of error = z��∕2 �� √ n = (2.576)(0.08165) = 0.2103 Step 4: Construct the confidence interval. The 99% confidence interval is ̄x − z��∕2 �� √ n < �� < ̄x + z��∕2 �� √ n 20.25 − 0.2103 < �� < 20.25 + 0.2103 20.04 < �� < 20.46 (rounded to two decimal places, like ̄x)
navidi_monk_elementary_statistics_2e_ch7-9
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