Consider a situation in which there is a population with a mean of 30 and a seco

Question

Consider a situation in which there is a population with a mean of 30 and a second population with a mean that differs. Both populations have a standard deviation of 4. If we took two samples of size 11 from each population, which of the values below is closest to the minimum magnitude of the difference between the means of the two populations that we would be able to detect with a two-tailed homoscedastic t test? 1.6 2.6 3.6 4.6 5.6 Consider a situation in which two populations differ by 20% and their standard deviations are approximately 30% of their means. If we take two identically sized samples from each population, which of the values below is closest to the minimum size of each sample we would need to take in order to correctly detect this difference with a two-tailed homoscedastic t test? 5 10 15 20 25

Explanation / Answer

Q10

Let X and Y represent the two populations. Assuming, X ~ N(µ1, 12) and Y ~ N(µ2, 22),

and given that both X and Y have the same standard deviation, i.e., 1 = 2 = , say,

the test statistic for H0: µ1 = µ2 is: t = (Xbar - Ybar)/{(2/n)}, where n = the common sample size. Under H0, t ~ t(2n - 2) and H0 is rejected if tobs > tcrit,/2 where = the specified level of significance, which we will take as 5% (since it is not specified in the question we will take the most frequently used 5%).

In the given question, n = 11, = 4 and finding from Standard Statistical Tables, t20, 0.025 = 2.086,

t = (Xbar - Ybar)/{4(2/11)} = (Xbar - Ybar)/2.4121.

We would be able to detect the difference (Xbar - Ybar) if (Xbar - Ybar)/2.4121 > 2.086 or

(Xbar - Ybar) > (2.4121 x 2.086) = 5.032 ANSWER     

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