Section 8.1 Confidence Intervals for a Population Mean, Standard Deviation Known 353 Procedure for Constructing a Confidence Interval for �� When �� Is Known Check to be sure the assumptions are satisfied. If they are, then proceed with the following steps. Step 1: Find the value of the point estimate ̄x, if it isn’t given. Step 2: Find the critical value z��∕2 corresponding to the desired confidence level from the last row of Table A.3, from Table A.2, or with technology. Step 3: Find the standard error ��∕ √ n, and multiply it by the critical value to obtain the margin of error z��∕2 �� √ n . Step 4: Use the point estimate and the margin of error to construct the confidence interval: Point estimate ± Margin of error ̄x ± z��∕2 �� √ n ̄x − z��∕2 �� √ n < �� < ̄x + z��∕2 �� √ n Step 5: Interpret the result. Rounding off the final result When constructing a confidence interval for a population mean, you may be given a value for ̄x, or you may be given the data and have to compute ̄x yourself. If you are given the value of ̄x, round the final result to the same number of decimal places as ̄x. If you are given data, then round ̄x and the final result to one more decimal place than is given in the data. Although you should round off your final answer, do not round off the calculations you make along the way. Doing so may affect the accuracy of your final answer. EXAMPLE 8.4 Construct a confidence interval The mean test score for a simple random sample of n = 100 students was ̄x = 67.30. The population standard deviation of test scores is �� = 15. Construct a 98% confidence interval for the population mean test score ��. Solution First we check the assumptions. The sample is a simple random sample, and the sample size is large (n > 30). The assumptions are met, so we may proceed. Step 1: Find the point estimate. The point estimate is the sample mean ̄x = 67.30. Step 2: Find the critical value z��∕2. The desired confidence level is 98%. We look on the last line of Table A.3 and find that the critical value is z��∕2 = 2.326. Step 3: Find the standard error and the margin of error. The standard error is �� √ n = 15 √ 100 = 1.5 We multiply the standard error by the critical value to obtain the margin of error: Margin of error = z��∕2 �� √ n = 2.326(1.5) = 3.489 Step 4: Construct the confidence interval. The 98% confidence interval is ̄x − z��∕2 �� √ n < �� < ̄x + z��∕2 �� √ n 67.30 − 3.489 < �� < 67.30 + 3.489 63.81 < �� < 70.79 (rounded to two decimal places, like ̄x)
navidi_monk_elementary_statistics_2e_ch7-9
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