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Princeton University Department of Operations Research and Financial Engineering ORF 245 Fundamentals of Engineering Final Exam

May 22, 2006 9:00 am-12:00 pm The Honor Pledge: ”I pledge my honor that I have not violated the Honor Code during this examination.”

Note: Please return this booklet along with your answer booklet.

1. (14 pts.) At a factory, motor-cycle tyre tubes are manufactured by an automatic process, and the production is so organized that the outgoing product includes, on average, 6% of defective tubes. The tubes are packed in batches of 24 each for dispatch to retailers.

(a) Calculate the probability that a randomly selected batch contains at least 2 de- fective tubes. (b) A retailer orders regularly 12 batches of tubes per month. If the manufacturer allows a refund of $1 on each defective tube received by the retailer, evaluate the expected amount of refund payable to the retailer in a month.

2. (14 pts.) A consultant surgeon lives on the outskirts of a city, and the direct route to the hospital passes through downtown. The journey by car takes, on average, 28 minutes with 7.5 minutes. The surgeon can bypass the congestion of downtown either by going through an adjoining village (average driving time of 32 min and SD of 4.3 min) or by taking the beltway around the city (average of 33 min and SD of 1.2 min). Assuming that all three times are normally distributed, determine which is the best route to take if the surgeon has

(a) 30 minutes; and (b) 35 minutes

to reach the hospital for an appointment. (Note that the best route for one case may not be the best route for the other case.)

1 3. (14 pts.) A DMV test center for learner drivers keeps track of the number of test appearances taken by an examinee for passing the driving test. Let X be a discrete denoting this variable and assume that it is distributed according to a zero-truncated Poisson distribution with the following p.m.f.:

e−1.861.86x P (X = x) = , x = 1, 2, ··· . (1 − e−1.86)x!

(a) Find the expected number of test appearances of an examinee for passing the driving test. Hint: you can use the known fact that the expected value of a Poisson distribution with parameter λ is λ. (b) Determine the conditional probability of a learner driver passing the test on the second attempt given that he/she has failed in the first examination.

4. (12 pts.) A city has decided to develop a computer graphics capability for its public works department. This capability includes the entry of the xy coordinates of utility line locations. This computer entry, called digitizing, was performed by two companies bidding for the digitizing contract. The accuracy of the entry process was mea- sured by the distance in meters between the actual and the recorded coordinate points for a sample of utility line locations for each company. The results of the measurement process are presented here:

Company AB x¯ 2.64 m 2.97 m s 0.63 m 0.85 m n 8 12

(a) Determine if company A is more accurate than company B. Do not assume that population are equal, and use a significance level of 5%. (b) Would the conclusion in (a) be any different had the population variances been assumed equal?

5. (18 pts.) The following figures, based on samples, relate to hair-color of girls in Edin- burgh and Glasgow:

City Of medium hair-color Sample size Edinburgh 4,008 9,743 Glasgow 17,529 39,764

Source: J.Gray(1907), Journal of the Royal Anthropological Institute, Vol.37.

(a) Estimate the difference between the true proportions of girls with medium hair- color in the two cities, and evaluate the of the estimate.

2 (b) Is it reasonable to say that there is a difference in the true proportion of girls with medium hair-color in the two cities? (c) By combining the two samples into a single one, determine the estimate of the true proportion of girls with medium hair-color in the two cities as a whole. Now, test the hypothesis that the true proportion of girls with medium hair-color in the combined population of the two cities is less than 45%.

6. (18 pts.) As part of an investigation into health service funding, a House subcommittee was concerned with the issue of whether mortality rates could be used to predict sickness rates. Data on standardized mortality rates and standardized sickness rates were collected for a sample of 10 regions and are shown in the table below:

Region Mortality rate x Sickness rate y (per 10,000) (per 1,000) 1 125.2 206.8 2 119.3 213.8 3 125.3 197.2 4 111.7 200.6 5 117.3 189.1 6 100.7 183.6 7 108.8 181.2 8 102.0 168.2 9 104.7 165.2 10 121.1 228.5

Data summaries:

x¯ = 113.61, Sxx = 780.709,y ¯ = 193.42, Syy = 3587.656, Sxy = 1278.118

(a) Calculate the correlation coefficient between the mortality rates and the sickness rates, and formulate the hypotheses and determine the P-value for testing whether the underlying correlation coefficient is zero against the alternative that it is positive. What assumption did you need to make in order to carry out this test? (b) Determine the fitted of sickness rate on mortality rate and test whether the underlying slope coefficient can be considered to be as large as 2.0. (c) For a region with mortality rate 115.0, estimate the sickness rate and calculate the 95% for this estimated rate.

7. (10 pts.) An to investigate the effects of a new technique for degumming of silk yarn was described in the article ”Some Studies in Degumming of Silk with Organic Acids” (J. Society of Dyers and Colourists, 1992: 79-86). One response variable of interest was y = weight loss (%). The experimenters made observations on weight loss for various values of three independent variables:

o • x1 = temperature ( C) = 90, 100, 110;

3 • x2 = time of treatment (min) = 30, 75, 120;

• x3 = tartaric acid concentration (g/L) = 0, 8, 16. In the regression analyses, the three values of each variable were coded as -1, 0, and 1, respectively, resulting in the following table:

Obs 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 x1 -1 -1 1 1 -1 -1 1 1 0 0 0 0 0 0 0 x2 -1 1 -1 1 0 0 0 0 -1 -1 1 1 0 0 0 x3 0 0 0 0 -1 1 -1 1 -1 1 -1 1 0 0 0 y 18.3 22.2 23.0 23.0 3.3 19.3 19.3 20.3 13.1 23.0 20.9 21.5 22.0 21.3 22.6

2 2 A quadratic regression model with k = 9 predictors - x1, x2, x3, x4 = x1 , x5 = x2 , 2 x6 = x3 , x7 = x1x2, x8 = x1x3, x9 = x2x3 - was fit to the data. Here are the results:

Regression Statistics R Square 0.93759 Standard Error 2.16237

ANOVA df SS MS F Regression 9 ? ? ? Residual 5 23.379 ? Total 14 374.636

Coefficient StdError t Stat P-value Lower 95% Upper 95% Intercept 21.967 1.248 17.595 0.000 18.757 25.176 x1 2.813 0.765 3.679 0.014 0.847 4.778 x2 1.275 0.765 1.668 0.156 -0.690 3.240 x3 3.438 0.765 4.496 0.006 1.472 5.403 x4 -2.208 1.125 -1.962 0.107 -5.101 0.684 x5 1.867 1.125 1.659 0.158 -1.026 4.759 x6 -4.208 1.125 -3.740 0.013 -7.101 -1.316 x7 -0.975 1.081 -0.902 0.409 -3.754 1.804 x8 -3.750 1.081 -3.468 0.018 -6.529 -0.971 x9 -2.325 1.081 -2.150 0.084 -5.104 0.454

(a) Does this model specify a useful relationship? State and test the appropriate hypotheses using a significance level of 0.01. Hint: complete the ANOVA table above in order to estimate the value of the relevant .

4 (b) Is there a useful relationship between weight loss and at least one of the second- order predictors x4, x5, ..., x9, after accounting for x1, x2, and x3? State and test the appropriate hypotheses at the 1% significance level. In order to answer this item, note that fitting a model with only x1, x2, and x3 as predictors produced the following results: Regression Statistics R Square 0.45596 Standard Error 4.30453

Coefficient StdError t Stat P-value Lower 95% Upper 95% Intercept 19.54 1.111 17.581 0.000 17.094 21.986 x1 2.813 1.522 1.848 0.092 -0.537 6.162 x2 1.275 1.522 0.838 0.420 -2.075 4.625 x3 3.438 1.522 2.259 0.045 0.088 6.787

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