Recall the mtcars data we used in our class early in the semester, then we used for constructing confidence interval for two population means difference,and, on the practice final exam. The data were "extracted from the 1974 Motor Trend US magazine, and comprise fuel consumption and 10 aspects of automobile design and performance for 32 automobiles (1973-74 models)." The horsepower of the engine measures its power output, and is found in the hp data column. A manual transmission requires the driver to manually shift the car into different gears, while an automatic transmission does so automatically. The horsepower for manual and automatic cars is not normally distributed, so like Homework 5, we will consider the natural logarithm of each. We are interested in testing whether the mean log-horsepower of manual transmission cars exceeds (is greater than) that of automatic transmission cars. Before we answer this question, we must determine whether the variances are equal or not. a) Please clearly write down appropriate hypothesis to test whether the variances of horsepower of manual and automatic cars are equal. Let = 0.05. b) You may assume that the distributions of log-horsepower for manual and automatic cars are normal. Use appropriate R function to decide whether the variances of log-horsepower of manual and automatic cars are equal with significance level = 0.05. Explain your test result statistically, and, in a scientific context. c) Now we revisit whether the mean log-horsepower of manual transmission cars exceeds that of automatic transmission cars. Please write down hypotheses clearly for this situation. d) What is pooled variance? What is its purpose, when is it used, and how is it calculated?

e) Use appropriate R function to test your hypotheses from part c. Give your conclusion in statistical and scientific terms. f) Suppose in the t.test() function, we changed the argument mu from 0 to 3. Which of the following output values would change: t, df, p-value, alternative hypothesis, 95 percent confidence interval, and/or sample estimates?