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ESE 326 - Probability and Statistics for Engineering - Spring 2016 Review Problems 1. A test has been developed to detect a particular type of arthritis in individuals over 50 years old. From a national survey it is known that about 10% of the individuals in this age group suffer from this form of arthritis. The proposed test was given to individuals from this group with confirmed arthritic disease and a correct test result was recorded in 85% of the cases. When the test was administered to individuals of the same group who were known to be free of the disease, 4% were reported to have the disease. If a random individual age 50 or older is tested and the test comes back “positive”, what is the probability that this person has the disease? 2. A standard deck of cards consists of 52 cards, 4 each of the numbers 2 to 10, and 4 each of jacks, queens, kings and aces. In the game of ‘poker’ 5 cards are distributed randomly (i.e., the chances to get a specific card are all equal) from a standard deck. A ‘full house’ is a combination of cards that consists of three cards of one type and two cards of another type, e.g., three ‘queens’ and two 7’s. What is the probability of getting three ‘queens’ and two 7’s? What is the probability of getting a ‘full house’ ? 3. For a normal distribution X with mean µ and variance σ 2 , 15% of the values of X are less than 12 and 40% are greater than 16.2. Find µ and σ. 4. The probability density function of jointly continuous random variables X and Y is given by for 0 ≤ y ≤ x ≤ 1, 2(x + y) f (x, y) = . 0 otherwise Find P (X + Y ≤ 1). 5. Suppose that X and Y are jointly continuous random variables with density given by x+y for 0 < x < 1, 0 < y < 1 f (x, y) = . 0 otherwise Calculate the density function g of the product Z = XY . 6. A certain chemical process has a yield of 70%. A new process has been devised and it is claimed that it produces a higher yield than the original process 90% of the time. In order to test this statement the new process is run 60 times and the number X of times in which the yield exceeds 70% is recorded. (a) Set up the appropriate hypothesis H1 and alternative null hypothesis H0 for this test. What is the null value? (b) Let us agree to accept the claim if the number X recorded is at least 59. What is the type I error α for this test? (c) Suppose the true probability is in fact p = 0.95. Then what is the type II error β for this test? 7. Quality and reliability are becoming important aspects of computer hardware and software. Past experience shows that the probability of failure during the first 1000 hours of operation for 16-kbit dynamic RAMs produced in the United states is 0.2. It is hoped that new technology and stricter quality control have reduced this failure rate. To verify this contention, 20 systems will be monitored for 1000 hours and the number of failures will be recorded. Set up the appropriate null and alternative hypotheses to test this assertion and explain the practical consequences of making a type I and type II error. (b) If H0 is true and p = 0.2, what is the expected number of failures during the first 1000 hours in the 20 trials? (c) Let us agree to reject H0 in favor of H1 if the observed number of failures, X, is at most 1. What is the type I error α for this test? (d) Suppose that it is essential that the test be able to distinguish between a failure rate of 0.2 and a failure rate of 0.1. Find the probability that the test as designed will be able to do so. That is, find the probability β of a type II error if indeed p = 0.1. 8. The following data represent the fuel gas temperature in degrees Fahrenheit (x) and unit heat rate in Btus per kilowatt hour (y) for a combustion engine to be used in coal gasification. x 100 150 200 250 300 350 400 450 500 y 99.1 98.5 98.2 98.0 97.8 97.6 97.5 97.0 96.8 Use these data for the next three problems. (a) Estimate the regression line µ̂Y |x = β0 + β1 x. (b) Test the hypothesis H1 : β1 ≤ 0 at a 5% level of significance. (c) Calculate 95% confidence intervals for the true intercept β0 and the true slope β1 . 9. The following data represent the known weights of calcium oxide (CaO) from ten different samples and the corresponding weights determined by a standard chemical procedure. Treat the known weights as independent variable x and the weight determined by the chemical procedure as dependent variable y. x 3.0 7.0 11.5 15.0 19.0 24.0 30.0 35.0 39.0 39.0 y 2.7 6.8 11.1 14.6 18.8 23.5 29.7 34.5 38.4 38.5 (a) Find the linear regression µ̂Y |x line to estimate the average weight of CaO based on the known weight. (b) For x = 15 estimate the average µ̂Y |15 and give a 90% confidence interval for this average. (c) How does the confidence interval change if we use this value as an estimate not for the average, but for a single realization for x = 15?