Practise calculating one-sample and matched-pairs t-statistics, choosing n−1 degrees of freedom, and finding p-values from the stated alternative.
A top-100, 7.0-rated ten)nis pro wishes to compare a new racket against his current model. He is interested in whether his hitting speed with the new racket is different from that with his old racket. He strings the new racket with the same type of strings at 60 pounds tension that he uses on his old racket. From past testing, the tennis pro knows that the average forehand crosscourt volley with his old racket is 82 miles per hour (mph). On an indoor court, using a ball machine set at 70 mph, which is the same speed he had his old racket tested against, he takes 47 swings with the new racket. (Assume that swings are independent of each other.) An associate with a speed gun records an average of 83.5 mph with a standard deviation of 3.4 mph. At the 0.01 level of significance, complete the appropriate inference procedure to determine if there is convincing statistical evidence that his mean hitting speed with the new racket is different from that with his old racket.
Complete the inference procedure, including calculating the appropriate statistics.
Let μ be the true mean hitting speed with the new racket. Test H0:μ=82 against Ha:μ=82. The swings are independent, and n=47≥30, so the normality condition is satisfied by the central limit theorem. The test statistic is t=3.0246, with df=46 and p=0.00406.