Question

You are planning a study of attitudes to the length of jail sentences for homicide, using a scale running from –4 to +4, where 0 indicates a judgment that current sentences are about right. Previous research suggests that the population SD for the scale is 1.2. You plan to use a single sample and would like to be able to detect a true effect of 0.5 scale units, using α = .01. If you use N = 100, using the same scale to compare attitudes in two very different neighborhoods. You would like to be able to detect a difference of 0.3 scale units. Consider power and make recommendations.

This is an exercise problem in one of the textbook by Geoff Cummings, Understanding The New Statistics, this is the first problem of chapter 12. This is all the information that it has and I must come with an answer regarding power and recommendations.

Answer 1

vs

test statistic,

when n=100 it is given that Hence the test statistic would be

The pvalue for this test statistic value is 0.0062<0.01 Hence the above null hypothesis can be rejected at 1% level of significance and we conclude that the sample provide enough evidence to support the difference of 0.3 scale units to be significantly different.

Now we come to power of making such a decision. Suppose if the alternative value

As we say above rejection would happen when

ie

ie

ie

And power of the test is defined as the probability of rejecting the null hypothesis when it is false (accepting the alternative value

ie P(/ )

For different alternatives considered ie for different values of we get the power to be different. You can even plot the power function for the different alternative values of considered. and see when it achieves the maximum value.