Quandaries
and Queries 

Please help with this question. Ms. Lisa Monnin is the budget director for the New Process Company. She would like to compare the daily travel expenses for the sales staff and the audit staff. She collected the following sample information. At the .10 significance level, can she conclude that the mean daily expenses are greater for the sales staff than the audit staff? What is the pvalue? Sales ($) 131 135 146 165 136 142 Audit ($) 130 102 129 143 149 120 139 Having problems finding the pvalue & unsure of the formula. 

Hi Kathy, I am going to use Mu_{1} for the mean daily travel expenses for all the audit staff and M_{2} for the mean daily expenses for all the sales staff. The null and alternate hypotheses are
The corresponding sample means I will call X_bar_{1} and X_bar_{2}. I got
For small, independent samples like you have the expression that you use for the variance depends on the assumptions you are making. I am going to assume that your teacher and textbook have you pool the data to estimate the variance using From your data I got s_{p}^{2} = 14.29 and the tvalue is then
You were asked to test the hypothesis at a 10% level of significance so you can reject the null hypothesis and conclude that the alternate hypothesis is true if your computed tvalue, 1.41, is larger that the 10% critical number for the tdistribution with 11 degrees of freedom. My statistics tables give the 10% critical number for the tdistribution with 11 degrees of freedom to be 1.363 so you can conclude, at the 10% level of significance that the daily travel expenses for the sales staff are larger than the daily travel expenses for the audit staff. Finally the pvalue which was really your question. I don't think of a formula for the pvalue I think of a diagram. If the diagram below is the graph of the tdistribution with 11 degrees of freedom and the computed tvalue, 1.41, is located on the horizontal axis then the area of the region shaded red is the pvalue. If you want a formula it is Pr(t > 1.41).
I can't give you a value for this area, the pvalue, the best I can do is estimate it. From my table of the critical numbers of the tdistribution with 11 degrees of freedom I know that the 10% number is 1.363 and the 5% number is 1.796. Again I see diagrams of the tdistribution with 11 degrees of freedom. One for the 5% number
where the area of the region shaded green is 5%, and one for the 10% number
where the area of the region shaded blue is 10%. By looking at the three diagrams I see that the area of the red region is smaller than the area of the green region, and larger than the area of the blue region. The area of the red region is the pvalue of the test so I can say
Without better tables or a way to calculate the area under the tdistribution curve the best I can say is that the pvalue is between 5% and 10%. Harley 

