A researcher is evaluating the influence of a treatment using a sample selected

Question

A researcher is evaluating the influence of a treatment using a sample selected from a normally distributed population with a mean of u = 112 and a standard deviation of o = 33. The researcher expects a 22-point treatment effect and plans to use a one-tailed hypothesis test with alpha = .05. For all parts below, explain logic, show all formulas, include all steps and plug in the numbers (show all work).

a. Compute the power of the test if the researcher uses a sample of n = 33 individuals.

b. Compute the power of the test if the researcher uses a sample of n = 68 individuals.

Explanation / Answer

a.

The standard error is sM = s/Ön = 33/Ö33 = 5.745

Using =.05, the critical region sample means beyond z = 1.96 or -1.96

The critical boundary of z = 1.96 corresponding to a location that is above µ=112 by a distance equal to 1.96sM = 11.26

Thus the critical boundary is located at a sample mean of M = 112+11.26 = 123.26.

In this case, the z-score is

               z = (M- µ)/ sM = (123.26 – 144)/5.745 = -3.088

Now, P (z >-3.088) = 0.99

So, the power of the test is 0.99.

b.

The standard error is sM = s/Ön = 33/Ö68 = 4

Using =.05, the critical region sample means beyond z = 1.96 or -1.96

The critical boundary of z = 1.96 corresponding to a location that is above µ=112 by a distance equal to 1.96sM = 7.84

Thus the critical boundary is located at a sample mean of M = 112+7.84 = 119.84.

In this case, the z-score is

               z = (M- µ)/ sM = (119.84 – 144)/4 = -6.04

Now, P (z >-6.04) = 0.9999

So, the power of the test is 0.9999.

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