End of Semester Extra Credit Problems (5 points each) 1. Each year QSR magazine does a survey of the drive-thru service of fast food chains. In a recent survey a sample of 375 McDonald's Restaurants found an average service time (order to pick up) of 174 seconds with a standard deviation of about 40 seconds. a) Find an 80% confidence interval for the average time of service at a McDonald's drive-thru. b) McDonald's has about 13,000 restaurants nationwide, the most of any chain. In the survey, about the same number of Taco Bell restaurants were visited even though they have far fewer outlets. Would the confidence interval for Taco Bell be wider, narrower, or about the same as the interval for McDonald's? 2. A genetic model says that a certain heart disease in dogs will occur in 25% of the puppies born to a mother with the disease (independently). Is the model correct? Suppose we examine the genetic model by testing 100 puppies from litters of dogs with the disease. What are the null and alternative hypotheses? 3. A researcher is interested in whether birth order has an effect on the size of a child's vocabulary. A vocabulary test is given to 100 children, all age five and all the oldest in their families. Later, the same vocabulary test is administered to the 100 next oldest sibling of these children when they reach five years of age (so both children in each family take the same test at age five). In this study, the oldest child scored higher on the vocabulary test for 57 of the 100 pairs of children (while the younger child scored higher for 43 of the pairs). Is this strong evidence that order of birth has an effect on vocabulary size, or can these data be explained by chance? Carry out the appropriate hypothesis test. Be sure to write down the null and alternative hypotheses, find the test statistic and the p-value, and state your conclusions. 4. A fast-food chain wishes to investigate the after cooking weight of their \"Quarter-Pound\" hamburger. The variation in the size of the hamburger patties is known to give a standard deviation of 0.015 pounds. However the machine that presses the patties may be improperly calibrated and produce patties that are systematically too large or too small. The company hires an independent laboratory to purchase 80 hamburgers at random times and independently weigh them using a procedure that is practically free of bias. The average of these weights is 0.245. Could the true average after-cooking weight of all of the "QuarterPound" hamburgers produced by this pressing machine still be 0.25 pounds, or is this strong evidence that the pressing machine is improperly set and the true average is not 0.25 pounds? Translate this into a statistical hypothesis-testing problem and carry out the test. Be sure to give the null and alternative hypotheses, the value of the test statistic, the p-value, and your conclusions. If your results were statistically significant, comment on whether you think they were also of practical significance. If your results were not statistically significant, comment on whether this experiment might have failed to detect a difference of practical significance. 5. Twenty percent of the cars on the road in one state are four-wheel drive. What is the chance that at least 22% of a group of 400 randomly selected cars from that state have fourwheel drive? Is it okay to use the normal approximation here? If no, say why. If Yes - calculate the probability. 6. The weights of 148 sets of twins born at the MetroHealth Medical Center in Cleveland, Ohio were recorded for a full year. How strongly is the weight of the first born associated with the weight of the twin? A scatterplot is shown below (all weights are in kilograms). a. The correlation between the weight of the first born and the weight of the twin is about A. -0.87 B. -0.36 C. 0.36 D. 0.87 Explain your choice. b. If the weights of the first born had been measured in pounds instead of kilograms, then: A. the value of the median weight of the first born twin would __________. B. the value of the standard deviation of the weights of the first born would ________. C. the value of the correlation between the two twins' weights would __________. Fill in the blanks from the possible choices listed below (Note. There are about 2.2 pounds in one kilogram). You may use an answer more than once. (1) be multiplied by 2.2 (2) be divided by 2.2 (3) stay the same (4) be multiplied by 2.2 times the correlation 7. The diameter of the moon is independently measured four times by a process that is free of bias. The individual measurements come out 2157, 2166, 2162, and 2155 miles, which average out to 2160 miles. One more measurement is about to be taken using the same process. When compared with the estimate of 2160 miles, you would expect this next measurement to be: A. more accurate as a measure of the true diameter of the moon. B. just as accurate at measuring the true diameter of the moon. C. less accurate as a measure of the true diameter of the moon. Pick one and explain. 8. A researcher is interested in the number of hours parents of pre-school children in the United States spend reading to their children. To investigate this, he obtains the student list from a local preschool and contacts the parents of 30 children from this list. Based on the parents' reports he found an average of 8.2 hours per week with a margin of error of 1 hour. a. What is the population of interest in this problem? b. What is the parameter of interest? c. Would you expect the parents' reports to generally underestimate or overestimate the amount of time they actually spend reading to their children? Explain. 9. Does the cocoa butter in chocolate raise serum cholesterol levels? Explain how you would design an experiment to answer this question. You may assume that 200 volunteers have agreed to digest pills containing cocoa butter on a daily basis for two weeks. 10. Thirty percent of the customers at a hardware store pay using a credit card. Consider the following two probabilities: Probability I - When 100 customers are selected at random - the probability that between 20 and 40 of them will use a credit card. Probability II - When 10 customers are selected at random - the probability that between 2 and 4 of them will use a credit card. Which is true? Pick one and explain briefly. A. The probability described in I is larger. B. The probability described in II is larger. C. The probabilities described in I and II are the same