Question

Question 1: (2, 2, 2 marks) A random sample of size 64 is taken from a normal population with u = 51.4 and 0 = 6.8. What is the probability that the mean of the sample will (a) exceed 52.9; (b) fall between 50.5 and 52.3; (c) be less than 50.6? Question 2:02, 2 marks) The actual proportion of families in a certain city who own, rather than rent, their home is 0.70. If 84 families in this city are interviewed at random and their responses to the question of whether they own their home are looked upon as values of independent random variables having identical Bernoulli distributions with the parameter = 0.70, with what probability can we assert that the value we obtain for the sample proportion will fall between 0.64 and 0.76, using : (a) Chebyshev's theorem; (b) the central limit theorem? Question 3: (2, 2, 2 marks) a. Find the sampling distributions of Y1 and Yn for random samples of size n from a continuous uniform population with a = 0 and B = 1. b. Find the probability that in a random sample of size n = 4 from the continuous uniform population above, the smallest value will be at least 0.20.

Answer 1

( 1 )

Given that M: 51.4,5 =68, 0=64 (a) PCX J52-9) = P(Z > 6.8/564 I = P(Z3 1.7647) = P(Z = 52.9-51.4) | (6) Plsos ex - 52.3) sos-s1.4 52.8 - L z 6.8/tou. 15-8/164 = P(-1.0588 < z < 1.6471) = P(Z < 1.6471) -P(Z <- 1.0588) = 0.9502 -0.1448 -10.8054 (c) P(Z <50.6) = P(2 . 50.6-51.4 6.81564 - Plze-0.9412)