A certain marathon has had a wheelchair division since 1977. An interested fan wondered who is faster: the men's marathon winn
er or the women's wheelchair marathon winner, on average. A paired t-test was performed on data from a random selection of 15 of the marathons to determine if there was evidence to indicate that the women's winning wheelchair time is faster than the men's winning running time, on average. What must be true about the population of differences in the women's wheelchair winning times and men's winning times at this marathon for conclusions from the paired t-test to be valid? Choose the correct answer below. A. The distribution of sample means of the differences will be approximately normal if there are at least 30 years of data in the sample and/or if the population of differences in winning times for all years is normal. B. Because there were at least 5 years of observations, the distribution of sample means of the differences will be approximately normal by the Central Limit Theorem. C. Because the sample size is large enough, the distribution of differences for all years will be normal. D. Because of the small sample size of differences in winning times between the women's wheelchair winner and the men's running winner, the distribution of sample means of the differences cannot be normal.
A. The distribution of sample means of the differences will be approximately normal if there are at least 30 years of data in the sample and/or if the population of differences in winning times for all years is normal.
Step-by-step explanation:
In other to perform a valid paired test, one of the conditions required is that, data for both groups must be approximately normal. To attain normality, the population distribution for the groups must be normal or based on the central limit theorem, the sample size must be large enough, usually n > 30. Hence, once either of the two conditions are met, the paired sample will be valid.
