Consider the question of whether sick days in a company are equally likely to occur Monday through Friday. 

a. State the null and alternative hypotheses using symbols. 

b. If we conduct a chi-square goodness-of-fit test, what are the degrees of freedom for this chi-square test? 

c. In the Goodness of Fit applet, enter some fake sample data (50 observations) for each of the five days of the week. It’s easiest to use a format like shown and press Use Table.

Your choice of observed counts is not important in this problem! All you are really doing is telling the applet the number of categories and the sample size. Use the Statistic pull-down menu to select the chi-square statistic. Then check Show Sampling Options and enter the hypothesized probabilities.   

Now generate 10,000 samples. In the null distribution, assuming 0.05 as the level of significance, use trial and error to determine a reasonable rejection region (to the nearest hundredth). What is the approximate probability of your making a Type I error? Include a screen capture.

d. Suppose the actual probabilities were 0.26, 0.16, 0.16, 0.16, 0.26. Check the “Alternative” box and enter these alternative probabilities. Draw a random sample. Which distribution (the null or the alternative) do these observations seem to be coming from? 

e. Generate 10,000 samples and then count how many are in the rejection region from (c). This estimates the power of the test. Include a screen capture. 

f. Write a sentence describing what the proportion from (e) means in the context of the problem. 

g. Repeat (c) and (e), changing (and documenting) the appropriate settings, to find the power for the following alternative probabilities: 0.32, 0.12, 0.12, 0.12, 0.32. Should the power be larger or smaller? Why? Is it? 

h. Repeat (c) and (e) for the MAD statistic. Is the power for the MAD statistic larger or smaller than for the chi-squared statistic? 

i. Repeat (h) for the Max-Min statistic. Which of the three statistics seems to have the least power? Does this make sense? Explain.

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