A car manufacturer claims that its vehicles average at least 25 miles per gallon, with a population standard deviation of σ=4. We take a sample of n=36 cars to test fuel efficiency, and with to performance a one-sided hypothesis test of the manufacturer’s claim at a level of α=.025. a. What is the critical value for the sample mean? b. Suppose the answer to part a is 23 (it is not, but assume it is to do part b). If the true value of efficiency in the population is 22.75, what is the power of this hypothesis test? 6.Suppose there are two types of Penn State Sports fans: Rabid (R), and Casual (R’). These fans either view a big game Live (L) or at Home (L’). Suppose 15% of Penn State fans are both Casual and watch the games at Home. If a fan watches at Home, the probability that he/she is Casual is 80%. Finally, suppose 10% of fans are Rabid. a. What is the probability of any fan watching the game at Home? b. If a fan watches the game Live, what is the probability that they are Rabid? 7.Suppose a random sample of size 21 is taken, and the variance of the sample is calculated to be 5. a. What is (approximately) the probability that the population variance σ² is less than 4? b. There is a 10% chance the population variance σ² will be greater than ___. 8.There is a wide disparity between education systems across the world. Suppose we have a dataset for 42 countries. The variable X is the average years of formal education for a person in a country, and Y is the per capita income (in thousands of dollars) in that country. We find a positive relationship between the two variables. Additionally, our regression shows the following results: a. According to the regression results above, what percentage of the variation in income is explained by the education level? b. What is the value of b1 in the regression Y=b0+b1X? (Hint: what is SSR and how is that related to b1?) c. Suppose the answer to part b is b1=4 (it is not, but to avoid compounding any mistakes, assume it is to answer the following). What is the value of b0? d. Continue to assume that b1=4. What is the 90% confidence interval for the true population slope coefficient β1?