Q. Suppose you have large samples and from two independent populations with mean
ID: 3072805 • Letter: Q
Question
Q. Suppose you have large samples and from two independent populations with means 1 and 2 each with finite variance but you don't know what the two distributions are (they are not necessarily normally distributed), and the two distributions could be different from each other. Use the central limit theorem applied to the two-sided 95% confidence interval for -N2. Finally, apply your confidence interval estimate to 15 observations from population 1 with a sample mean of 60 and- 100 and 12 observations from population 2 with a sample mean of 55 and = 121 . Compare your results to that of the usual equal-variance t-based confidence interval for the difference between means of two normally distributed random variables.Explanation / Answer
Pooled Variance
sp^2 = ((df1)(s1^2) + (df2)(s2^2)) / (df1 + df2) = 301051 / 25 = 12042.04
Standard Error
s(M1 - M2) = ((sp^2/n1) + (sp^2/n2)) = ((12042.04/15) + (12042.04/12)) = 42.5
Confidence Interval
1 - 2 = (M1 - M2) ± ts(M1 - M2) = 5 ± (2.06 * 42.5) = 5 ± 87.53
1 - 2 = (M1 - M2) = 5, 95% CI [-82.53, 92.53].
You can be 95% confident that the difference between your two population means (1 - 2) lies between -82.53 and 92.53.
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