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362 PART V: Analyzing and Reporting ResearchWe have already described procedures for computing the sample mean ( __ X ) andthe estimated standard error of the mean ( s __ X ). The unfamiliar symbols in thetwo equations for the limits of the 95% confidence interval are t and .05.We briefly discussed the alpha () level of .05 in Chapter 6. It is typicallyassociated with inferential tests of statistical significance (i.e., NHST), and wewill have much more to say about alpha levels in Chapter 12. In the case of confidenceintervals, (1 level of confidence) is expressed as a proportion. So,for the 95% confidence interval, (1 .95) .05 and for the 99% confidenceinterval, (1 .99) .01.The t statistic included in the equation is defined by the number of degreesof freedom, and the statistical significance of t can be determined by lookingin Appendix Table A.2. For a single sample mean, the degrees of freedom areN 1. You will learn more about the t statistic in Chapter 12 when we discussNHST. At this point let us simply concentrate on the calculation and properinterpretation of a confidence interval using the above formulas.An example will illustrate how we obtain a confidence interval for a singlemean. Suppose you obtained a random sample of students at your universityand measured their intelligence using a brief but valid and reliable measure ofthis construct. Assume 30 students (N 30) were tested and the mean intelligencescore was 115 with a sample standard deviation of 14. The population ofstudents is represented by the thousands of students attending your university.And while the sample mean is a good point estimate of the population mean(i.e., our best guess of the population mean), we must acknowledge that if anotherrandom sample of 30 students were selected and tested the sample meanwould not likely be exactly 115. There will be some slippage, or “error,” due tothis random process. Recall that the standard error of the mean is one measureof the error in estimation.Rather than rely simply on a point estimate of the population mean, wecan obtain an interval estimate by finding the 95% confidence interval for thepopulation mean using the formulas presented earlier. We first calculate theestimated standard error of the mean:__s X ____ s _____ 14 N 30 ____ 145.48 2.55Next, we obtain the critical t value. Because there were 30 students, the degreesof freedom associated with the t statistic are 30 1 or 29. Using Table A.2 wecan find that the value of t with alpha of .05 and 29 degrees of freedom is 2.04.Using the formulas for the confidence interval, we haveUpper limit of 95% confidence interval 115 [2.04][2.55]Lower limit of 95% confidence interval 115 [2.04][2.55]Upper limit 115 5.20 120.20Lower limit 115 5.20 109.80We may state that there is a .95 probability that the interval 109.80 to 120.20 contains(“has captured”) the population mean (see Box 11.3).