National Motors has equipped the ZX-900 with a new disk brake system. We define the stopping distance for a ZX-900 to be the distance (in feet) required to bring the automobile to a complete stop from a speed of 35 mph under normal driving conditions using this new brake system. In addition, we define m to be the mean stopping distance of all ZX-900s. One of the ZX-900’s major competitors is advertised to achieve a mean stopping distance of 60 feet. National Motors would like to claim in a new advertising campaign that the ZX-900 achieves a shorter mean stopping distance. Suppose that National Motors randomly selects a sample of 81 ZX-900s. The company records the stopping distance of each automobile and calculates the mean and standard deviation of the sample of 81 stopping distances to be 57.8 ft and 6.02 ft., respectively.
a. Calculate a 95 percent confidence interval for μ. Can National Motors be 95 percent confident that μ is less than 60 ft? Explain.
b. Using the sample of 81 stopping distances as a preliminary sample, find the sample size necessary to make National Motors 95 percent confident that x-bar is within a margin of error of one foot of μ.