## CEE 260 / MIE 273 Probability & Statistics Name: Final Exam, version D — 100 points (120 minutes) PLEASE READ QUESTIONS CAREFULLY and SHOW YOUR WORK! CALCULATORS PERMITTED – ABSOLUTELY NO REFERENCES! 1. Suppose the waiting time (in minutes) for your 911 SC Targa to reach operating temperature in the morning is uniformly distributed on [0,10], while the waiting time in the evening is uniformly distributed on [0,5] independent of morning waiting time. a. (5%) If you drive your Targa each morning and evening for a week (5 morning and 5 evening rides), what is the variance of your total waiting time? b. (5%) What is the expected value of the difference between morning and evening waiting time on a given day? 2. (10%) Find the maximum likelihood estimator (MLE) of ϴ when Xi ~ Exponential(ϴ) and you have observed X1, X2, X3, …, Xn. 2 3. The waiting time for delivery of a new Porsche 911 Carrera at the local dealership is distributed exponentially with a population mean of 3.55 months and population standard deviation of 1.1 months. Recently, in an effort to reduce the waiting time, the dealership has experimented with an online ordering system. A sample of 100 customers during a recent sales promotion generated a mean waiting time of 3.18 months using the new system. Assume that the population standard deviation of the waiting time has not changed from 1.1 months. (hint: the source distribution is irrelevant, but its parameters are relevant) a. (15%) What is the probability that the average wait time is between 3.2 and 6.4 months? (hint: draw a sketch for full credit) b. (10%) At the 0.05 level of significance, using the critical values approach to hypothesis testing, is there evidence that the population mean waiting time to accept delivery is less than 3.55 months? c. (10%) At the 0.01 level of significance, using the p-value approach to hypothesis testing, is there evidence that the population mean waiting time to accept delivery is less than 3.55 months? 3 4. Porsche AG is a leading manufacturer of performance automobiles. The 911 Carrera model, Porsche’s premier sports car, reaches a top track speed of 180 miles per hour. Engineers claim the new advanced technology 911 GT2 automatically adjusts its top speed depending on the weather conditions. Suppose that in an effort to test this claim, Porsche selects a few 911 GT2 models to test drive on the company track in Stuttgart, Germany. The average top speed for the sample of 25 test drives is 182.36 mph, with a standard deviation of 7.24 mph. a. (5%) Without using complete sentences, what might be some problems with the sampling conducted above? Identify and explain at least 2. b. (15%) Using the critical values approach to hypothesis testing and a 0.10 level of significance, is there evidence that the mean top track speed is different for the 911 GT2? (hint: state the null and alternative hypotheses, draw a sketch, and show your work for full credit) c. (10%) Set up a 95% confidence interval estimate of the population mean top speed of the 911 GT2. d. (5%) Compare the results of (b) and (c). What conclusions do you reach about the top speed of the new 911 GT2? 4 5. (10%) Porsche USA believes that sales of the venerable 911 Carrera are a function of annual income (in thousands of dollars) and a risk tolerance index of the potential buyer. Determine the regression equation and provide a succinct analysis of Porsche’s conjecture using the following Excel results. SUMMARY OUTPUT Regression Stat istics Multiple R 0.805073 R Square 0.648142 Adjusted R Square 0.606747 Standard Error 7.76312 Observations 20 ANOVA df SS MS F Significance F Regression 2 1887.227445 943.6137225 15.65747206 0.000139355 Residual 17 1024.522555 60.26603265 Total 19 2911.75 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept 23.50557 6.845545641 3.433702952 0.003167982 9.062731576 37.94840898 Income 0.613408 0.125421229 4.890786567 0.000137795 0.348792801 0.878024121 Risk Index -0.00126 0.004519817 -0.278357691 0.784095184 -0.010794106 0.008277854 BONUS (5 points) What is the probability that 2 or more students in our class of 22 have the same birthday?

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