Question

1. The FBI compiles data on robbery and property crimes and publishes the information. A ran- dom sample of pickpocket offenses yielded the losses, in dollars, described by the following summary statistics: * = 513.32, s = 262.23, and n = 20. We want to use this data to make inference on u, the mean loss in all pickpocket offenses in the U.S. Assume a = 0.05 and the data comes from a normal distribution with unknown variance, answer the questions below: I (a) Construct a CI for p. (b) If we were to test Ho : u = 398 vs. Hq : u * 398, would we reject or accept the null? Briefly explain your answer. (c) Suppose you are told that the population standard deviation is 258. What test statistic would you use now to test the hypotheses in the previous part? Perform the test. Does it contradict the answer in the previous part? (d) Using the population standard deviation, and suppose the true mean is 402, what is the probability of Type II error for the hypothesis test in part (b)?

Answer 1

given, do. Os W = 513.32, S = 262,23, n=20 (a) ch for u use t-score since n (=200 230 degree of freedom = ni-19. al2 = 0.025 turi, ale = 20093. ă-t x . &t there on the C513.32 - 2.093 x 262.23 513.324 20434 262,23 2o ป.6 636.04) (1) CI =) (utarat sa in 1 (390.5a

We (b Hoil = 398 Haiu 739 8. compute the t-score, as t= Roll = 1.966 Score 7 votre less than the t-value for 9s1. confidence interval, we fail to reject the null hypothesis accept mull hypothesis use 2-score now (6 given 8= 258 Z-Scre = x-u olun 258/20 513.32-398 - 1.998 Also 2-value for 95% confidence two-tailed = 1.96

95% confidence (0.05) 2-value (1.96), reject the 2) It contradicts the answer, 2-score (1.998) greater than the mell hypothesis pw, (d) given all = 402 true mean 258, Ho Mo=398 P (type Il error) = P (accept mull & null is false) accept nullo if è lesbo (Mo-24 x 0 , lot Zal, (398 - 1.96 x 258 398+1,96 x 259 Vio Szo ( P (accept 258/520 258/850 = (284,92 ,51607). will 402) = P ( X 8 284,42 4 X251607/4-402 P(2>(204.92 -.402), 24/51.07-402 = P(23-2002, 221.89) = P( 241.89) - PC2L-2002) = (1-0.0294) - 08217 (B) =0.9489 type llerra error (B