2 Condence Intervals and Proportions Scenario 1: Suppose you are working for a certain candidate in an election. The candidate claims that 70% of voters support him. Can you imagine anyone that hold? 5. You get a random sample of 7,500 voters with sample proportion of 3'5 2 0.72 voters that support the candidate. Test the null hypothesis that p = 0.7 at the 5% level. Does it make sense to say you have \"proven\" or \"accepted" the null hypothesis? (7 pts) 6. Compute the 95% condence interval for p using the information from the above question. What does this interval say? (7 pts) 7. Using the information from the above two questions, does it necessarily make sense to the candidate that he is \"wrong\"? (7 pts) Scenario 2: The manager of a bank claims that approximately 15% of their loans resulted in default. Let F = 1 if the loan resulted in default, and F = 0 otherwise. Then, clearly the distribution of F is: F ~ Bernoulli(0.15) Suppose that in a typical year, the bank grants 100 loans to businesses. Assume that defaults are i.i.d. events (that is, if two firms are granted loans, whether the first firm defaults is independent of whether the second firm defaults). 8. Let K be the number of loans granted in a typical year that will eventually end up in default. What is the distribution of K? (7 pts) 9. Looking at a random sample of 100 loans, an audit finds that only 7 of those loans ended up in default. Determine a 95% confidence interval for p. (7 pts) 10. Perform a hypothesis test on the manager's claim. Was the claim accurate? What is the new distribution of K? (7 pts) 33 One-sided Binomial Confidence Intervals Scenario 1: We are analyzing 0-1 (binary or dummy) survey data with sample size n = 13. Define the random variable Xi = 1 if the person is satisfied, and zero if not where X; ~ Bernoulli(p) and each is i.i.d. across all i = 1, 2, ..., 13 people surveyed. Let Y be the distribution (sum) of the n i.i.d. Bernoulli random variables. Accordingly, Y ~ Binomial(n, p). 11. Suppose all 13 people surveyed were satisfied. Construct a 95% confidence interval for the unknown p without using the CLT. What is the smallest resonable value of p? Should our 95% confidence interval include p = 0.83? (Note that you just need to use the CI bounds to find the answer). (7 pts) 12. Recall that from class we can verify the result to the above question by using the binomial probability of all successes out of n trials in the form Pr(Y = n) = p". Applying this logic, check your answer to question 11. (7 pts) 13. Consider now building a confidence interval for p, but under the assumption that all 13 people in our surveyed sample were NOT satisfied. Construct a 95% confidence interval for p without using the CLT. What is the largest reasonable value of p? Should our 95% confidence interval include p = 0.27? Use the CI bounds to determine the answer, and double check with the binomial probability of all failures out of n trials Pr(Y = 0) = (1 -p)". (7 pts)Scenario 2: I have just challenged golfer Tiger Woods (widely regarded to be the best golfer ever) to a putting contest. We are each putting the same 10-foot putt. Dene the random variable 1'} = 1 if Tiger makes his i-th putt and 0 if he misses. Similarly. dene L5 = 1 if I make my i-th putt and 0 otherwise. Accordingly. each putt is i.i.d. such that: T.- ~ Bernoullirr) and L; w Bernoulli(p1,) Tiger and I are each going to hit 10 putts, and whoever makes the most wins. 14. Consider building a condence interval for pg. The putting green was a little rough, and the following data was recorded for my 10 putts: {0I 0,0, 0, 0, 0. 0. 0. 0.0} So, I missed all 10 of my putts (] still think there was sabotage from Tiger), which means that 151. 0. Construct a 95% condence interval for the true 13;, without using the Central Limit Theorem. What is the largest value of pi, you would call \"reasonable\"? (7 pts) 15. Tiger is great on the golf course, but he says that he's not the best putter {at least not the best 10-foot putter). In fact, he claims that he only makes 1 out of every 4 putts. That is, his true proportion of putts is pr = 0.25. To make myself feel better about my lousy day. I am going to test whether 1 am truly as good as Tiger by considering the same proportion for myself In = 0.25. Help me in answering the following two questions: (a) Should my 95% condence interval include 151, = 0.25? (7 pts) (b) Check your answer to the above question by calculating PrLL = 0) in a sample of size n = 10. Did you get the same answer? (7 pts) Assignment 2 1 Independence and Identical Distributions Consider a. scenario where we are working with two random variables R and Q where R ~ Bernoulli and Q is not Bernoulli. Assume the marginal distribution Pr(Q = 1) = 0.50 and the conditional distribution Pr(R = 1 | Q : 1) = 0.50 is symmetric to the conditional distribution Pr(R = 0 I Q = 1) = 0.50. FinallyI assume that PT(R : 1) = 0.53. 1. Complete the joint probability table below (it helps to begin with the shaded box, i.e. the joint probability PT(R : 1 D Q : 1) given the above information). (7 pts) Q 1 2 Pr(R) R D 1 0.58 Pr(Q) 0.50 2. R m Bernoulli(0.42). (3 pts) (a) True (b) False 3. (R | Q : 1) ~ Bernoulli(0.5). (3 pts) (a) Time (h) False 4. The random variables R and Q are: (3 pts) (a) i.i.d. (b) identically distributed but not independent (c) independent but not identically distributed (d) none of the above