URGENT

35. Consider all courtroom trials with a single defendant who is charged with a felony. Suppose that you are given the following probabilities for this situation. Eighty-two percent of the defendants are, in fact, guilty. Given that the defendant is guilty, there is a 75 percent chance the jury will convict the person. Given that the defendant is not guilty, there is a 40 percent chance the jury will convict the person. For simplicity, assume that the only options available to the jury are: to convict or to release the defendant.

(a) What proportion of the defendants will be convicted by the jury?

(b) Given that a defendant is convicted, what is the probability the person is, in fact, guilty? (c) What is the probability that the jury will correct decision?

36. Consider all courtroom trials with a single defendant who is charged with a felony. Suppose that you are given the following probabilities for this situation. Seventy-five percent of the defendants are, in fact, guilty. Given that the defendant is guilty, there is a 70 percent chance the jury will convict the person. Given that the defendant is not guilty, there is a 40 percent chance the jury will convict the person. For simplicity, assume that the only options available to the jury are: to convict or to release the defendant.

(a) What proportion of the defendants will be convicted by the jury?

(b) Given that a defendant is convicted, what is the probability the person is, in fact, guilty?

(c) What is the probability that the jury will make a correct decision? (

d) Given that the jury makes an incorrect decision, what is the probability that the decision is to release a guilty person?

37. Recall that a confidence interval is too small if the number being estimated is larger than every number in the confidence interval. Similarly, a confidence interval is too large if the number being estimated is smaller than every number in the confidence interval. Each of the four researchers selects a random sample from the same population. Each researcher calculates a confidence interval for the median of the population. The intervals are below. [24, 41], [30, 39], [20, 33], and [35, 45].

(a) Nature announces, "Two of the intervals are correct and two are too small." Given this information, what is the narrowest interval that is known to contain the median? (Hint: The answer is not any of the four confidence intervals.)

(b) Nature announces, "Two of the intervals are correct, one interval is too small and one interval is too large." Given this information, what is the narrowest interval that is known to contain the median? (Hint: The answer is not any of the four confidence intervals.)

(c) It is possible all of these intervals are incorrect. For example, if = 100 then every interval is incorrect. But what is the maximum number of these intervals that can be correct? What values of will give this maximum number of correct intervals? (Hint: The answer is not any of the four confidence intervals.)