Starbucks has 15, 328 stores worldwide. (This information is not current.) Let's suppose 6, 284 of those stores are run by women. A statistician monitoring the sales of a particular product noticed that the top-selling stores for this product tended to be run by women. In particular, there were 3, 832 stores in the top 25%, and 1, 620 of those stores were run by women.

1. (a)Use counting rules to compute the probability that 1, 620 of the top 3, 832 stores are run by women when the gender identity of the person running the store is not relevant to sales.
2. (b)Monte Carlo sampling with a computer can be useful in two ways: (1) it can be used to prove when your formulas are wrong and lend empirical support when they are right, and (2) it can be used to do calculations when they are tedious or impossible by hand. Use R's sample() (or your favorite language) to repeat this sales experiment 10, 000 times under the assumption of no bias toward women. Turn in your code.
3. What proportion of time do you see exactly 1,620 female-run stores in a random selection of 3, 832 stores? Do you believe it? Why or why not?
4. What proportion of time do you see 1, 620 female-run stores in a random selection of 3, 832 stores? Is there evidence that women have some kind of advantage for selling this product?
5. (c)Write the formula in Part (a) for a general question where there are N stores, K of which are women-run, n in the top 25% and k of these women-run. This is the pmf of a hypergeometric random variable. Use R to compute the probability of Subpart (ii) of Part (b). Is it close to your previous calculations by Monte Carlo?
6. (d)Show the derivation for the expectation of a hypergeometric random variable (try it before you mine it, because it is doable), showing your work. What is the expectation for the Starbucks data?

(e) Using limiting arguments discussed in class, show that the hypergeometric converges to the Bin(n, p) distribution as N, K with p = K/N .