Section 8.2 Confidence Intervals for a Population Mean, Standard Deviation Unknown 369 Checking the assumptions When the sample size is small (n ≤ 30), we must check to determine whether the sample comes from a population that is approximately normal. This can be done using the methods described in Section 7.6. A simple method is to draw a dotplot or boxplot of the sample. If there are no outliers, and if the sample is not strongly skewed, then it is reasonable to construct a confidence interval using the Student’s t distribution. Check Your Understanding 1. Use Table A.3 to find the critical value t��∕2 needed to construct a confidence interval of the given level with the given sample size: a. Level 95%, sample size 15 2.145 b. Level 99%, sample size 22 2.831 c. Level 90%, sample size 63 1.671 d. Level 95%, sample size 2 12.706 2. In each of the following situations, state whether the methods of this section should be used to construct a confidence interval for the population mean. Assume that �� is unknown. a. A simple random sample of size 8 is drawn from a distribution that is approximately normal. Yes b. A simple random sample of size 15 is drawn from a distribution that is not close to normal. No c. A simple random sample of size 150 is drawn from a distribution that is not close to normal. Yes d. A nonrandom sample is drawn. No Answers are on page 378. Objective 2 Construct confidence intervals for a population mean when the population standard deviation Is unknown Constructing a Confidence Interval for �� When �� Is Unknown The ingredients for a confidence interval for a population mean �� when √ �� is unknown are the point estimate ̄x, the critical value t��∕2, and the standard error s∕ n. The margin √ n. When the assumptions for the Student’s t distribution are met, we of error is t��∕2s∕ can use the following step-by-step procedure for constructing a confidence interval for a population mean. Procedure for Constructing a Confidence Interval for �� When �� Is Unknown Check to be sure the assumptions are satisfied. If they are, then proceed with the following steps. Step 1: Compute the sample mean ̄x and sample standard deviation s, if they are not given. Step 2: Find the number of degrees of freedom n − 1 and the critical value t��∕2. Step 3: Compute the standard error s∕ √ n and multiply it by the critical value to obtain the margin of error t��∕2 s√ n . Step 4: Use the point estimate and the margin of error to construct the confidence interval: Point estimate ± Margin of error ̄x ± t��∕2 s√ n ̄x − t��∕2 s√ n < �� < ̄x + t��∕2 s√ n Step 5: Interpret the result.
navidi_monk_elementary_statistics_2e_ch7-9
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