Stat1600 Midterm #2 Solution to Form A

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1 Stat1600 Midterm #2 Solution to Form A 1. Of 100 adults selected randomly from one town, 64 have health insurance. A researcher wants to construct a 95% confidence interval for the percentage of all adults in the town who have health insurance. (a) (10 points) What is the estimate of the true proportion of adults with health insurance? ANSWER: 64% CALCULATION/ (show your work) ˆp successes sample sie or 64%. (b) (10 points) What is the standard error of the estimate of the proportion? ANSWER: 4.8% CALCULATION/ (show your work) SE ˆp(1 ˆp) n 0.64(1 0.64) or 4.8%. (c) (10 points) Calculate the 95% confidence interval for the population proportion (in percentages)? ANSWER: (54.6%,73.4%) CALCULATION/ ( }{{}, ) (0.546, 0.734) (54.6%, 73.4%). ME

2 2. A company estimates that there is approximately an 8% chance of an incoming shipment of parts containing defects. Suppose the company ordered 75 shipments last month. (a) (10 points) It is known that John inspected 10 shipments. What is the chance that he found exactly 3 shipments containing defects? ANSWER: The chance that John found exactly 3 shipments containing defects is P (X 3) 10! 3!7! (.08)3 (1.08) 7 120(.08) 3 (.92) ! ! (.08)3 (.92) 7 (b) (10 points) The number of shipments containing defects in last month is expected to be around, give or take. ANSWER: around 6, give or take 2.35 µ n p σ n p q (c) (10 points) What is the probability that at most 4 shipments in last month containing defects? (Hint: Draw rectangles representing the area of interest (4 or less) and use the normal approximation.) ANSWER:

3 P (X 4) P (X < 4.5) (see figure below) ( P Z < ) P (Z < 0.64) 2.35 P (Z > 0.64) 1 P (Z 0.64) Suppose the state health agency wants to compare motor vehicle injury rates when a car seat belt is used and when it is not used. Data from the department of transportation is shown on the table below. Injury? Sample Seat belt Used? Yes No Sie No Yes (a) (10 points) Estimate the difference in the proportions of injuries between those who did not use seat belt and those who used seat belt. Answer:

4 Note that the respective sample sies were added to the table above. Consider those who did not use seat belt as group 1 and those who used seat belt as group 2. Then the difference in the proportions, p 1 p 2, of injuries between those who did not use seat belt and those who used seat belt is estimated by ˆp 1 ˆp (b) (10 points) Estimate the odds ratio of having an injury when a seat belt is not used and when it s used. Answer: ÔR (c) (10 points) Find a 95% confidence interval for the true odds ratio. Answer: (1.235,3.659) ln(ôr) ln(2.126) SE ME 1.96SE % c.i. for ln(or): ( , ) (0.2113, ). 95% c.i. for OR:(e , e ) (1.235, 3.659). (d) (10 points) Interpret the result you found in part (c). The odds of injuries if seat belt is not used is between to times as large as the odds of injuries if seat belt is used. The comparison is significant since the interval excludes (5 points) Extra Credit Problem Assume that a statistical study was performed, and it was found that there was a strong association between ice cream sales and drowning deaths. In other words, for a given day, the more ice cream that was sold the more drowning deaths there were. Obviously, if we simply consider this association as an indicator of cause-and-effect, we would conclude that 4

5 we must ban ice cream sales in order to reduce the number of drowning deaths. Which of the following statements best describes a fair assessment (from a statistical point of view) of this study? (a) Any observational study, like this one, definitely shows cause-and-effect. We cant actually ban ice cream sales, but we should certainly communicate these results so the public is informed. (b) This is actually a good study. If we COULD ban ice cream sales,then we should try to. We dont want people drowning. (c) As in all observational studies, this outcome certainly shows cause-and-effect. (d) There could be a confounding variable, perhaps temperature, that is related to both ice cream sales and drowning deaths. We really cant show cause-and-effect here. Answer: (d) 5

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