# On a given day stock market prices respond to

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On a given day, stock market prices respond to information about the general economy and therefore, it may be expected that the returns for an individual company may be correlated with the overall performance of the market. That is, a non-zero covariance between the x and y observations is realistic. The task of interest is to obtain a confidence interval estimate for the difference between the mean return for the company and the mean return for the market portfolio. The differences are generated as: i i i y x d - = for i = 1, 2, . . . , 20 The sample mean and standard deviation of the differences were calculated as: 0.173 - = d and 1.391 = d s The negative value for d says that the sample mean for the company returns is less than the sample mean for the market returns. This does not imply that this is the case for the population – the population means are unknown.

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Econ 325 – Chapter 8 5 For the purpose of the exercise, set the confidence level to 0.95. A 95% interval estimate is: n s t d d c ± An illustration of the t-distribution critical value c t is below. PDF of ) 19 ( t tc 0 -tc Area = 0.95 Upper Tail Area = 0.05 / 2 = 0.025 Lower Tail Area = 0.025 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: 2.093 = c t Econ 325 – Chapter 8 6 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: 20 1.391 2.093 0.173 ± - This gives the lower and upper limits: [ 0.48 0.82 , - ] head2right Note that the interval contains the value zero (the lower limit is negative and the upper limit is positive). This suggests the possibility that 0 Y X = μ - μ or Y X μ = μ .
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