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 p. 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 n = 81 ZX-900s. The company records the stopping distance of each automobile and calculates the mean and standard deviation of the sample of n = 81 stopping distances to be  = 57.8 ft and s = 6.02 ft.
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 n = 81 stopping distances as a preliminary sample, find the sample size necessary to make National Motors 95 percent confident that x is within a margin of error of one foot of μ.