Q1

We wish to estimate the differenceABin mean fuel efficiencies for twobrands of automobile. We measure the fuel efficiencies (in mpg)of samples of cars for each brand. The data are shown below with some summary statistics:

Brand A 40.7 43.8 36.5 38.4 45.0 37.6 41.2 mean = 40.46 std. dev. = 3.17 Brand B 37.4 33.9 40.1 38.9 36.2 mean = 37.30 std. dev. = 2.41

Assume that fuel efficiencies are known to follow normal distributions for both brands. (a) What is the estimate of the common variance of fuel efficiencies for the two brands of automobile?(Round your answer tothreedecimal places.) (b)Calculate a99% confidence interval for the difference in the mean fuel efficiency forthe two brands of automobile.

We would like to conduct a test ofH0:A=Bvs.Ha:ABat the 1% level of significance to determine whether the true mean fuel efficiency for Brand A is different than that for Brand B. (c) What is the value of the test statistic for the appropriate test of significance? (d) What is the P-value of the test? (e) What is the correct conclusion of the test? (f) Suppose we had instead conducted the test using an analysis of variance. What would be the value of the test statistic and the P-value?

(g)Verify your results by conducting theappropriate hypothesis test inJMP. To do this, create a column calledFuel Efficiencyand enter all 12 data values in this column (all7 data values for Brand Aand then all5 data values for Brand B). Create a second column calledBrandand typeAin the first7 rows (beside the data values for the Brand A cars) andBin the next5 rows (beside the data values for the Brand B cars). Go toAnalyze > Fit Y by X. ChooseFuel Efficiencyas Y andBrandas X. ClickOK. Now, if you want to conduct a pooled two-sample t test, under the red arrow, selectMeans, Anova, Pooled t. If you want to conduct a conservative (unpooled) two sample t test, under the red arrow, selectt Test.

What is the exact P-value of the test?

(h) Suppose we had instead used the critical value method to conduct the test. What would be the decision rule and the conclusion?

Q2

*Give all answers totwodecimal places. We would like to conduct an analysis of variance at the 10% level of significance to compare the residential energy consumption of households in four regions of the U.S. We measure the energy consumption in a sample of 12 homes in the Northeast Region (NE), 9 homes in the Northwest Region (NW), 16 homes in the Southeast Region (SE)and 17 homes in the Southwest Region (SW).

Suppose energy consumption are known to follow a normal distribution with common standard deviation for each of the four regions. The ANOVA table (with some values missing) is shown below:

Source of Variation	df	Sum of Squares	Mean Square	F
Groups			872.01
Error
Total		12100.4

(a) What are the hypotheses for the appropriate test of significance?

(b) Find all missing values in the table above. Show your calculations and display the final complete ANOVA table. (c) What is the P-value for the appropriate test of significance? (d) What is the correct conclusion of the test? (e) Suppose you had used the critical value method to conduct the test. What would be the decision rule and the conclusion? (f)What is the estimate of the common population standard deviation? (g) The sample means and standard deviations for the Northeast Region and Northwest Region are, respectively,xNE=56.20,sNE=12.61,xNW=48.73,sNW=11.18. Calculate a 90% confidence interval forNENW,the differencein the mean energy consumption of Northeast Region and Northwest Region.