After studying how Michael Phelps swims his races, we wanted to see if times actually went down. In a study of 40 swimmers the sample mean was 2.10 seconds of improvement with a standard deviation of 8.15. At 95% confidence the t-score is 1.685. What can we conclude about the null hypothesis?
Explanation / Answer
Given n = 40 sample size x1 - x2 = 2.10 seconds Improvement in swimming times of the sample s = 8.15 seconds sample standard deviation t-score = 1.685 = 0.05 significance level Null and alternate hypothesis can be Ho : 1 = 2 1, 2 are the means of swimming times before and after studying Michael Phelps swimming Ha : 1 > 2 Indicating swimming times have gone down after study We first find the degrees of freedom. Since we use the sample readings before and after the study, we consider two samples of size 40 each Degrees of Freedom = (40 - 1) + (40 - 1) =78 We find the p-value for the t-score = 1.685 with degrees of freedom = 78 p-value = 0.048 Since p-value(0.048) < (0.05), we reject the null hypothesis and accept the alternate hypothesis Conclusion : There is sufficient statistical evidence to show there is improvement in the swimming times after stydying how Michael Phelps swims races.