171
FINAL Math EXAM
Spring 06
Directions: You may use two pages (two sides of a sheet of paper) of notes. Calculators are permitted and we also provide you with tables. All work must be shown in the answer booklet. Be sure it is complete, neat, and in order. Clearly indicate your final answers. Proper mathematical justification must be provided in order to earn full credit; correct answers with no work shown may receive no credit. Note that using the calculator does not take the place of showing your work. You may use standard notation; any new notation or abbreviations you use must be clearly defined. Note that the test totals 150 points. You have 2 1 2 hours for this exam. Good luck! 1. [22 points] Insurance. Based on past experience, an insurance Number of company expects its automobile policyholders to have 0, 1, or 2 accidents Probability accidents per year with probabilities indicated in the table. 0 0.7 a) Find the mean and standard deviation of the random variable X 1 0.2 describing the number of accidents per year for these auto 2 0.1 policyholders. [5] b) Suppose the company has 200 automobile policyholders, and that their accident experiences are independent of each other. What is the probability that the mean number of auto accidents the company experiences in a year will be greater than or equal to 0.45 per policyholder? [4] c) Calculate the mean and standard deviation of the total number of auto accidents per year the company should expect their 200 customers to have. [4] d) The company also has a large number of motorcycle policyholders, and expects them to have a total of 60 motorcycle accidents per year, with a standard deviation of 5 accidents. What is the probability that the company will face more motorcycle accidents than auto accidents this year? [5] e) Have you used the Central Limit Theorem in any of the above parts? If so, explain how it was used. [4] 2. [20 points] Sampling. The accompanying page of software output shows 12 histograms of sample statistics calculated from many random samples taken from two different populations. The histograms in one column show estimates of a population proportion; in the other column they show estimates of a population mean. Within each column the different histograms reflect estimates based on samples of different sizes. a) Which column (Left or Right) contains histograms that display sample proportions? How can you tell? [3] b) Based on the column of histograms that you believe display estimates of a proportion: i) arrange the histograms (by letter) in order of increasing sample size; [3] ii) estimate the value of p for the population. [2] c) Based on the column of histograms that you believe display estimates of a mean: i) arrange the histograms (by letter) in order of increasing sample size; [3] ii) estimate the value of for the population; [2] iii) estimate the value of ! for the population given that the largest sample size used was n = 1024. [4] d) Briefly explain the reasoning you used to sort the histograms by sample size. [3]
3.
[24 points] Inference. Read each scenario and indicate: (1) which inference procedure would be appropriate (one proportion, 2 proportions, one mean, two means, matched pairs, goodness of fit, homogeneity, independence, or regression slope); (2) the proper test statistic (z, t, or ! 2 ); and (3) the number of degrees of freedom (if appropriate). NOTE: Dont actually do the inference!! a) An insurance company compiled data on the number of cars stolen per 1000 residents for 23 cities. The company also determined the number of police officers per 1000 residents for each of the cities in order to see if there was any indication that a higher police presence might deter auto theft. [4] b) It certainly costs more to attend Cornells College of Arts and Sciences than the College of Agriculture and Life Sciences. How does this impact the amount of financial aid students receive? Data reporting dollar amounts of financial aid were collected from random samples of 50 A&S students and 50 CALS students in order to create a confidence interval for the difference in aid received. [4] c) Data on electricity usage by a random sample of over 3000 homeowners showed that 63% were within one standard deviation of the mean, 91% within two standard deviations, and 97% within three standard deviations of the mean. Do these observations suggest that home electricity consumption may not be normally distributed? [4] d) Years ago it was unusual for a high school student to apply to more than a few colleges, but the expanded use of the internet and the common application seems to have contributed to an increase lately. We plan to survey a random sample of 80 high school seniors in order to estimate of the typical number of applications students now submit. [4] e) Last years auto industry sales figures indicate that 22% of the new cars sold were SUVs. A recent poll, taken after gas prices hit $3 per gallon, asked 650 people who were shopping for a new car if they were planning to buy an SUV; only 116 said yes. Does this indicate a significant change in consumer sentiment? [4] f) A university agronomist has developed a new variety of oats and wants to see if it will produce a better yield than the variety currently grown in his state. Twenty nearby farmers offer to participate in a field test. Because crop yield is affected by soil type, the researcher randomly selects a field on each farm and then has the farmer plant the usual kind of oats on one half and the new variety on the other half. When the oats are harvested the farmers will report the yield (in bushels per acre) of each variety of oats. [4]
4. [15 points] Stress. A poll described in the October 23, 2000 issue of Newsweek asked 245 mothers and 161 fathers of children aged 0-3 about loss of sleep as a source of stress. Results showed that 41% of the mothers and 36% of fathers reported that they experienced this problem. a) Give a 90% confidence interval for the difference between the proportions of mothers and fathers who suffer stress from loss of sleep. (Be sure to discuss the appropriate conditions.) [8] b) Explain how your result from part a) can be used to test (at = ! 0.10 ) whether or not theres a difference in the proportions of mothers and fathers who experience sleep-loss stress. (Your explanation should also include statements of your hypotheses and your conclusion in context.) [7]
5. [14 points] Greener cars. The EPA is thinking of raising fuel economy requirements for passenger cars to 30 mpg. Engineers for one auto company have developed an engine modification that they believe will surpass this high efficiency goal. Understandably, company executives need a demonstration that the modification will work before theyll be willing to make a costly change in the companys engine manufacturing plant, so the engineers plan to install prototypes on 20 cars. Some company employees will then drive these cars for a month of commuting, keeping careful records of their gas mileage. After collecting all the data, the engineers will test a hypothesis about the new engines at ! = 0.05 . a) What null hypothesis will the engineers test? [2] b) Whats their alternative hypothesis? [2] c) Explain in this context what a Type I error would be, and describe the risk such an error poses for the company. [3] d) Explain in this context what a Type II error would be, and describe the risk such an error poses for the company. [3] e) Given the importance of this research, company executives direct the engineers to increase the power of the test they plan to conduct. Describe two ways they could accomplish this, and explain a disadvantage of each approach. [4] 6. [13 points] Pulse rates. 46 students in an introductory Statistics class were asked to measure their resting pulse rates (in beats per minute). Then everyone ran in place for one minute and afterwards measured their pulse rates again. Using the summary statistics shown on the accompanying page of software output, create a 90% confidence interval for the amount of the increase. (Be sure to discuss the appropriate conditions and to interpret your interval in the proper context.) 7. [18 points] Mummies. The frequencies of blood type for different races are generally different from each other, especially for populations with little intermixing. For example, consider Egyptian rulers. An interesting question is whether the rulers circa 1700 BC and the rulers circa 1200 BC were from the same race. One Observed Era theory is that in 1700 BC, the rulers might have been Blood Circa Circa the foreign Hyksos, who were a reproductively isolated type 1700BC 1200BC Total population. It turns out that one can determine the O 20 26 46 blood type of mummies (who were all in the ruling A 23 12 35 class in Egypt). Data from research done on samples of B 17 7 24 Total 60 45 105 mummies from each era are summarized in the first table shown. a) Is this a goodness-of-fit test, test for homogeneity, or a test of independence? [2] b) Formulate appropriate hypotheses. [2] c) This second table shows all the expected counts. Expected Era Show how to calculate the expected count of 26.3 in Blood Circa Circa the first cell. [2] type 1700BC 1200BC Total d) Discuss the conditions or assumptions needed in O 26.3 19.7 46 order to test your hypothesis. [2] A 20.0 15.0 35 e) Test your hypothesis and state your conclusions (in B 13.7 10.3 24 context, of course). [10] Total 60 45 105
8. [24 points] Housing prices. The accompanying page of software output displays the results of a Data Desk analysis of the relationship of housing price (in thousands of dollars) to area (square footage) for 106 housing sales. The data are a random sample of moderately priced homes in Albuquerque that were resold during 1993. Shown are the regression table, scatterplot, residuals plot, a histogram of the residuals, and summary statistics for e...