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Unformatted text preview: e level to 0.95. A 95% interval estimate is: d ± tc sd n An illustration of the t‐distribution critical value t c is below. PDF of t (19 ) Area = 0.95 Lower Tail Area = 0.025 Upper Tail Area = 0.05 / 2 = 0.025 -tc 0 tc To look‐up the critical value from the Appendix Table for the t‐distribution select the degrees of freedom (n−1) = 19, and set the upper tail area to 0.025. To check the method, with Microsoft Excel select Insert Function: TINV(0.05, 19) This returns the answer: 5 t c = 2.093 Chapter 9 By using the numerical results, the 95% interval estimate for the difference in population mean returns for the company and the market is calculated as: − 0.173 ± 2.093 ⋅ 1.391 20 This gives the lower and upper limits: [ − 0.82 , 0.48 ] Note that the interval contains the value zero (the lower limit is negative and the upper limit is positive). This suggests the possibility that μ X − μ Y = 0 or μ X = μ Y . That is, from the sample of data, there is evidence that the two population means are the same. 6 Chapter 9 Chapter 9.2 More Confidence Intervals for the Difference Between Two Population Means Another problem of interest is to compare the population means of two independent samples. Consider two independent random samples from normal populations: • the first sample has n x observations from a population with mean μ X . An estimator of the population mean is the sample mean X ....
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