Spring 2016 INEN 4320: Statistical Decision Making - Midterm Exam Name:_________________________ Signature ______________________ By signing the honor pledge you are swearing that you have neither given nor received any unauthorized assistance before or during this examination. Any form of cheating will not be tolerated. 1. (5 points) A synthetic fiber used in manufacturing carpet has tensile strength that is normally distributed with mean 75.5 psi and standard deviation 3.5 psi. Find the probability that a random sample of n = 6 fiber specimens will have sample mean tensile strength that exceeds 75.75 psi. 2. (5 points) Consider the probability density function 1 ( ) = 2 , 0 < , 0 < < Find the maximum likelihood estimator for . 3. (15 points) Consider the probability function () = 0.5(1 + ), 1 1 (a) Find the moment estimator for . (8 points) (b) Show that = 3 is an unbiased estimator for . (3 points) (c) Find the maximum likelihood estimator for . (4 points) 4. (5 points) A random sample has been taken from a normal distribution and the following confidence intervals constructed using the same data: (38.53, 48.87) and (36.59, 50.81) (a) What is the value of the sample? (2 points) (b) One of these intervals is 99% CI and the other is a 95% CI. Which one is the 95% CI and why? (3 points) 5. (15 points) A research measured the weight of 30 rats under experiment controls. Suppose there are 6 underweight rats. (a) Calculate a 95% two-sided confidence interval on the true proportion of rats that would show underweight from the experiment. (8 points) (b) Using the point estimate of p obtained from the preliminary sample, what sample size is needed to be 95% confident that the error in estimating the true value of p is less than 0.02. (5 points) (c) How large must the sample be if we wish to be at least 95% confident that the error in estimating p is less than 0.02, regardless of the true value of p? (2 points) Page 1 of 2 6. (35 points) A consumer product firm operates packaging lines that fill bottles of different sizes and contents. Machine A is dedicated to fill bottles with one liter and it is known to be a normally distributed process with standard deviation of 5 ml. (1 liter= 1000 ml.). A sample of size 10 is taken and the average is computed to be 998.4 ml per bottle. (a) With the information provided by the sample, find the 95% confidence interval for the mean. (5 points) (b) Which is the minimum required sample size, for the length of the 95% confidence interval of the mean to be 1.0 ml. or less? (The confidence interval length = 2 x Error) (5 points) (c) Using the sample mentioned above, test the null hypothesis that bottles are filled with mean of 998 ml with significance level of 5%. The alternative hypothesis is that the bottles are not filled on average with 998 ml. What is your conclusion? (9 points) (d) Find the P-value of the result of this test of hypothesis with this sample. (5 points) (e) Determine the probability of error type II with this hypothesis, if the true value of the mean is 996 ml. (6 points) (f) If the true mean of this process is 1000 ml, with standard deviation of 5 ml, what is the proportion of bottles that are filled from 998 ml to 1006 ml? (5 points) 7. (20 points) Consider the hypothesis test 0 : 1 = 2 against 1 : 1 < 2 . Suppose that sample size n1 = 15 and n2 = 15, that 1 = 6.2 2 = 7.8, and 12 = 4 22 = 6.25. Assume that 12 = 22 . The data are drawn from normal distribution. Use significance level = 0.05. (a) Test the hypothesis. (15 points) (b) Find the P-value. (5 points) Page 2 of 2