Question: Let Xi,X2, . . . Xn be independent and identically distributed (lid) draws from some distribution. You wish to estimate the population median 7]. Recall

Let Xi,X2, . . . Xn be independent and identically distributed

Let Xi,X2, . . . Xn be independent and identically distributed (lid) draws from some distribution. You wish to estimate the population median 7]. Recall that the median is defined so that 17(7)) : 0.5. You wish to test H0 : n S 14 vs. H1 : 7] > 14. It can be shown that an appropriate test statistic is the number of observations with a value greater than 14. In other words, let Z be our test statistic where: _ 0 Xi14 K'{1 X9141 and Z : 2:121 K- a. What is the probability that K; = 1? (use the definition of the median) (int) b. What is the distribution of your test statistic Z? Hint: it is not normal, but it is another very common distribution you have seen many times. Recall that you have iid data. (2pt) c. Intuitively explain why your rejection region will be ofthe form C = {Z 2 c}. (1pt) d. Find c for a sample size n = 6 and an o: no greater than 0.05. What is your actual significance level? (3pt) e. Given the data a: : {4, 8, 15, 16, 23, 42} calculate your test statistic, state your conclusion, and find your p-value. (3pt)

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