# Which distribution to use in the following scenario?

Cross Validated Asked by Arindam Bose on December 3, 2020

In a random sample of 86 athletes participating in the 2012 Summer Olympics, 63 were found to have sponsorship by private companies. Based on this sample, conduct a hypothesis test to see if there evidence to support the claim that the true proportion (among all athletes participating in the 2012 Summer Olympics) of those with private company sponsorship is greater than 70%. Which distribution is used to compute the p‐value, and what range does the p‐value lie in?

a. Z distribution, p < 0.5
b. t distribution, p < 0.5

I believe that the statistic that needs to be used is the t stat, since we have a sample of the overall population and we are deriving the population sd from the sample sd. But the official answer stated is that a Z distribution is supposed to be used.

Sporty question. Why not use a binomial test for this purpose? You want to test whether your

$$H_1: P > 0.70$$

as opposed to the opposite

$$H_0: P leq 0.70$$

The binomial test reasons on a fraction in a distribution, and it is exact.

Reference

S. Siegel, N.J. Castellan. Nonparametric statistics for the behavirioral sciences, McGraw-Hill, 1988, chapter 4.

Correct answer by Match Maker EE on December 3, 2020