The Trial Urban District Assessment (TUDA) measures educational progress within

Question

The Trial Urban District Assessment (TUDA) measures educational progress within participating large urban districts. TUDA gives a reading test scored from 0 to 500. A score of 243 is a "basic" reading level for eighth-graders. Suppose scores on the TUDA reading test for eighth-graders in your district follow a Normal distribution with standard deviation sigma = 55. In 2013 the mean score for eighth-graders in your district was 255. You plan to give the reading test to a random sample of 25 eighth-graders in your district this year to test whether the mean score mu for all eighth-graders in your district is still above the basic level. You will therefore test the following. H_0: mu = 243 H_a: mu > 243 If the true mean score is again 255, on average students are performing above the basic level. You learn that the power of your test at the 5% significance level against the alternative mu = 255 is 0.29. (a) You could get higher power against the same alternative with the same a by changing the number of measurements you make. Should you make more measurements or fewer to increase power? We should make more measurements. The answer depends on the measurements. We should make fewer measurements. The number of measurements does not affect the power of the test. (b) If you decide to use a = 0.10 in place of alpha = 0.05, with no other changes in the test, will the power increase or decrease? The answer depends on the measurements. The power will not change because it is not affected by the significance level. The power will increase. The power will decrease. (c) If you shift your interest to the alternative mu = 260 with no other changes, will the power increase or decrease? The power will increase, because this alternative is more likely to be detected. The power will decrease, because this alternative is more difficult to detect. The power will increase, because this alternative is more difficult to detect. The power will decrease, because this alternative is more likely to be detected. The power will not change, because it is not affected by the value of the alternative.

Explanation / Answer

(a) answer is we should make more measurements

Reason:

The power of a test = rejecting a false hypothesis

Our test statistic is given by

t = (255-243)/(55/sqrt25)

When n ( here 25) increases the value of test statistic increses leading to increase in probability of rejecting the null hypothesis .

Thus power increases with increase in sample size

(b) the power will decrease

Reason: as alfa decreases, rejection region decreases, that is probability of rejecting the null hypothesis decreases

(c) the power will increase as the alternative is more likely to be detected

Reason : as true mean score is increased the effect size ( true mean score- hypothetical mean score) increases , which leads to increse in probability of rejecting the null hypothesis, that is accepting the alternative hypothesis

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