Final Exam Blank Problem 1 A diagnostic test for a certain disease is said to be 90% accurate in that, if a person has the disease, the test will detect it with a probability of 0.9. Also, if a person does not have the disease, the test will report that he or she does not have it with a probability of 0.9. Only 1% of the population has the disease in question. If a person is chosen at random from the population and the diagnostic test indicates that she has the disease, what is the conditional probability that she does in fact have the disease? Final Exam Blank Problem 2 The number of messages posted in a bulletin board is a poisson random variable with a mean of 10 messages per hour. a) What is the probability that less than two messages are posted in one hour? b) Suppose that no message has been posted for 3 hours. Find the probability that another will elapse before the next message arrives. Final Exam Blank Problem 3 Hits on a high-volume website are assumed to follow a poison distribution with a mean of 10,000 hits per day. What is the probability of more than 10,150 hits in a day? Approximately the question by using a normal distribution. Final Exam Blank Problem 4 a) Determine the value of c that makes the function f(x,y)=c(x+y) a joint probability density function over the range 0<x<3, x<y<x+1. b) For the Joint Probability density function in a), find E[X]. Final Exam Blank Problem 5 a) A rivet is to be inserted into a hole. A random sample of n=18 is selected and the diameter of the hole is measured, which is assumed to be normally distributed. The sample variance of the diameter of these 18 samples is s2= (0.006)2 mm2. Construct a 95%^ 2 sided symmetrical confidence interval on the true variance of the diameter. b) The height of a certain group of adults is assumed to be normally distributed with an unknown mean μ centimeters and a known standard deviation σ=4 centimeters. The design of the experiment’s team has to recommend the sample size that is required to construct a 95% two sided symmetrical confidence interval on the mean μ, which has a total width of 3.0 centimeters, or the confidence interval as +/- 1.50 centimeters. Find the value of the required sample size for this study. Final Exam Blank Problem 6 The mean water temperature downstream from a power plant cooling tower discharge pipe should be no more than 1000F. It is assumed that the water temperature follows a normal distribution with parameters μ and σ. An environment engineer is interested in the following hypothesis: Ho: μ =100, H1: μ>100 Part A Past Experience indicates that the standard deviation σ of the temperature is 40F. The water temperature is measured on 7 randomly chosen days and the average temperature is found to be 102.10F. a) Should the water temperature be judged acceptable with a type 1 error, α =0.01? b) What is the P-Value for this Test? c) What is the probability of accepting the null hypothesis at α=0.01 if the true water temperature has a mean of 1080F? d) What sample size would be required to detect a true mean of 1020F if we wanted the power of the test to be at least 0.95? Final Exam Blank PART B a)The Standard deviation of the water temperature is not known. We have the following values of the water temperature on 7 randomly chosen days. 104, 96, 101, 99, 106, 103, 102 b) What is the range of the P-value for this test? Final Exam Blank ` A manufacturer of resistors is proud of the low proportion of defective (p) Resistors produced by the company. A customer decides to test the following hypothesis: H0: p=0.01, H1>0.01 For a statistical analysis, the customer takes a random sample of 150 resistors and finds that four (4) are defective. a) Would the customer reject the null hypothesis? Use α=0.05 b) What is the P-value for this hypothesis test?