1. The following data is obtained from a population with a known variance of 9. 9.81 1491 19.39 10.63 14.80 20.73 25.23 8.32 15.83 19.26 [f the data is known to come from a Gaussian population, find the 95% confidence interval for the mean. 2. A random sample of size 10 is taken from a Gaussian population with mean y; and variance g2 = 225 yield an average X; = 170.2. An independent sample of size 12, taken from a Gaussian population with mean u, and variance 0 = 256 yield an average X, = 176.7. Find the 95% confidence interval for p; . 3. Let Xy, X5, -, Xy be arandom sample of size N from a population having a Poisson distribution with unknown parameter u. Find an estimator for the parameter y based on (a) Maximum likelihood estimation. (b) Method of moments. It 1s desired to estimate the fraction 6 of defective computers in a production line. A random sample of size n is selected from the line and the number of defective ones is observed. For a single computer, let X be a random variable with value 1 if the computer is defective and 0 if the computer 1s not defective. Find an estimator for the parameter 8 using (a) Maximum likelihood estimation. (b) Method of moments. Past experience indicates that the time for computer science students to complete a quiz 1s a normal random variable with mean 35 minutes and a standard deviation of 4.3 minutes. If a random sample of 20 computer science students took an average of 33.1 minutes to complete the quiz, test the hypothesis at a 0.05 significance level that u = 35 minutes against the alternative that y 6): HO: g = 80 H]_: 0 = 91 (a) Assuming that the prior probabilities, P(H,) and P(H,) are equally likely, determine the statistical test to decide between H, and H;. (b) Find the power of the test