Mathematical and Computational Methods for Engineers E155C, Winter 2004 Review Problems 1. If a driver consistently uses motor oil A, then the probability of a motor repair in excess of $600 prior to 75,000 miles on the odometer is 0.03; however, for motor oil B, the probability is 0.08. In a test sample, 40 cars have been consistently lubricated with motor oil A and 60 with motor oil B. a) if a test car is selected at random, what is the probability that it will require $600 of repair work before being run for 75,000 miles ? b) given that a car requires motor work in excess of $600 prior to 75,000 miles, what is the probability it was lubricated with motor oil A ? 2. If X and Y are independent, show that the characteristic function for Z X Y can be expressed as follows: Z (w) X (w)Y (w) 3. If the average number of claims handled daily by an insurance company is 5, what proportion of days have less than 3 claims ? What is the probability that there will be 4 claims in exactly 3 of the next 5 days ? 4. A contractor purchases a shipment of 100 transistors. It is his policy to test 10 of these transistors and to keep the shipment only if at least 9 of the 10 are in working condition. If the shipment contains 20 defective transistors, what is the probability it will be kept ? 5. The joint density of X and Y is given by f ( x, y ) 2 over the interval 0 x y , 0 y 1 . Are X and Y independent ? 1 n ( X i X )2 n 1 i 1 estimator of the true variance 2 6. Show that s 2 is an unbiased 7. The lot average yield strength of a certain alloy is required to be not less than 150 ksi. The standard deviation based on past experience is 7 ksi and yield strength is known to possess a normal distribution. Ten specimens were drawn at random from a large lot and tested with the following results: 145 142 151 153 138 131 146 153 131 150 a) using a 5% level of significance, should the lot be accepted ? b) what is the probability of accepting a lot that has a lot average yield of 145 ksi ? c) if standard deviation were not known, find a 95% confidence interval for the mean strength 8. A manufacturer claims that the resistance of carbon film resistors has a standard deviation of 0.50 Ohm. The sample variance of 20 resistors has been found to be equal to 0.49 Ohm. a) would you reject the manufacturer’s claim at the 5% level of significance ? b) what is the probability of rejecting the manufacturer’s claim when 1 1.5 0 ? 9. The data below show measurements of the corrosion effects for steel pipe with two kinds of coating: X 72 63 53 47 38 56 55 75 68 57 61 52 67 Y 69 69 58 52 33 57 54 66 72 71 69 53 60 a) if X and Y are both normally distributed with known common variance 2 9 , test the hypothesis at the 5% significance level that the kind of coating used has no effect on corrosion b) redo part a) if X and Y are both normally distributed, and 2 is known to be the same for both X and Y but its value is unknown c) redo part a) if X and Y are not normally distributed, i.e. use a non-parametric test 10. The following data gives the frequency distribution of demand for a certain product per day based on the records of 1000 days: Demand/day 0 1 2 Freq. 626 274 80 3 15 4 4 5 1 Fit a Poisson distribution to the above data and test for the goodness of fit at the 5% level of significance. 11. The following data represent the effect of annealing temperature x on the ductility of brass, measured in terms of elongation y: x, deg C y, (%) 300 400 500 600 700 800 40 50 55 60 67 70 a) determine the regression line of y on x b) estimate y when x 550 c) find a 95% confidence interval for the slope of the regression line 12. Suppose that 20 specimens of metal were tested to ascertain whether there is a relationship between yield and tensile strength. The estimate of the correlation coefficient was found to be 0.85. a) is there evidence of a statistically significant correlation between yield and tensile strength ? b) find a 95% confidence interval for using Fisher’s transformation