Question: U.S. Courts of Appeals judges hear cases in three judge panels. Suppose you want to investigate the differences in the wording of rulings between all male panels and panels that have at least one female judge.

Suppose also that there is a limit to how many cases you can investigate, because you want to be able to read each case in full. You decide to randomly sample 100 cases that were heard by an all-male panel and 100 cases that were heard by a panel with at least one female judge.

As an initial analysis, you want to make descriptive inferences about the number of times that the word "harassment" is used in each ruling in the full population of cases. We are interested in \({{\mu }_{f}}\) , the population mean number of times that the word "harassment" is used in cases with at least one female judge on the panel. We are also interested in \({{\mu }_{m}}\) , the population mean number of times that the word "harassment" is used in cases with an all-male panel.

In our random sample of panels with at least one female judge we have the following summary statistics: sample size (nf = 100), sample mean ( \({{\bar{x}}_{f}}\) = 9.22), and sample standard deviation (sf = 19.84).

In our random sample of panels with all male judges we have the following summary statistics: sample

size (nm = 100), sample mean ( \({{\bar{x}}_{m}}\) = 5.16), and sample standard deviation (sm = 14.77).

1. i. Form a large sample 90% confidence interval for \({{\mu }_{f}}\).
   ii. Provide an interpretation of this confidence interval in terms of the panels with at least one female judge.
   iv. Do the 90% intervals for \({{\mu }_{f}}\) and \({{\mu }_{m}}\) overlap? What does this imply about whether the 95% intervals overlap?
   v. Explain whether using t-quantiles instead of the large sample Z-quantiles would be justifiable from a theoretical perspective. Explain, whether it would matter from a practical perspective.
2. i) Calculate an estimate of \(V\left[ {{{\bar{X}}}_{f}}-{{{\bar{X}}}_{m}} \right]\)
   ii) Using \({{\bar{x}}_{m}}\) , \({{\bar{x}}_{f}}\) and your estimated variance, calculate a large sample 90% confidence interval for \({{\mu }_{f}}\) - \({{\mu }_{m}}\)
   iii) Using \({{\bar{x}}_{m}}\) , \({{\bar{x}}_{f}}\) and your estimated variance, calculate a large sample 89% confidence interval for \({{\mu }_{f}}\) - \({{\mu }_{m}}\)
   iv) What do the previous two items tell you about the p-value for the two sided test of Ho: \({{\mu }_{f}}\) - \({{\mu }_{m}}\) = 0?
   
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