Question: Manufacturers of two competing automobile models, Gofer and Diplomat, each claim to have the lowest mean fuel consumption. Let μ1 be the mean fuel consumption in miles per gallon (mpg) for the Gofer and μ2 the mean fuel consumption in mpg for the Diplomat. The two manufacturers have agreed to a test in which several cars of each model will be driven on a 100-mile test run. Then the fuel consumption, in mpg, will be calculated for each test run. The average of the mpg for all 100-mile test runs for each model gives the corresponding mean. Assume that for each model the gas mileages for the test runs are normally distributed with σ = 2 mpg. Note that each car is driven for one and only one 100-mile test run.
a. How many cars (i.e., sample size) for each model are required to estimate μ1 - μ2 with a 90% confidence level and with a margin of error of estimate of 1.5 mpg? Use the same number of cars (i.e., sample size) for each model.
b. If μ1 is actually 33 mpg and μ2 is actually 30 mpg, what is the probability that five cars for each model would yield x1¯ ≥ x2¯?