LAST NAME (Please Print): KEY FIRST NAME (Please Print): HONOR PLEDGE (Please Sign): Statistics 111 Midterm 1 This is a closed book exam. You may use your calculator and a single page of notes. The room is crowded. Please be careful to look only at your own exam. Report all numerical answers to at least two correct decimal places or (when appropriate) write them as a fraction. All question parts count for 1 point, unless otherwise indicated. 1
1. A retirement home has two television rooms. At 9 p.m. on Sunday, residents must choose which of two new shows to watch: Game of Cards or House of Thrones. Out of 50 people who watched Game of Cards, 40 liked it. Among the 50 who watched House of Thrones, only 20 liked it. The following Sunday, one of the television sets breaks down. The director decides that everyone should watch Game of Cards, since it was more popular. But Nurse Ratchet knows that of the audience for Game of Cards consisted of 45 Democrats and 5 Republicans, and that all 5 Republicans hated the show. And of the audience for House of Thrones, there were 10 Democrats and 40 Republicans, and 8 Democrats liked the show. The first step is to put the information into a table, such as the following. In the parentheses, I ve listed Democrats first, then Republicans: Like Not Like Game of Cards (40, 0) (5, 5) House of Thrones (8, 12) (2, 28) 0 What percentage of Republicans like Game of Cards? From the table, 0%. 30% What percentage of Republicans like House of Thrones? 12/(12+28), or 30%. 0.57 or 0.58 What is Nurse Ratchet s estimate of the proportion of people who will like House of Thrones? In terms of the Berkeley example, like/dislike is accept/reject, GoC/HoT is male/female, and R/D is like major. So the formula is proportion of Dems * prop. of Dems who like HoT + proportion of Reps * proportion of Reps who like HoT = (55/100) (8/10) + (45/100)(12/40) = 0.575. How does Nurse Ratchet explain the situation to the director? Democrats tend to like television shows, but Republicans don t. More Democrats saw GoC, so it seemed more popular, but that was misleading. 2
2. A high school has 20 each of freshmen, sophomores and juniors, and 10 seniors. The proportions of women in each group are 50%, 50%, 60% and 70%, respectively. Fresh Soph Jr Sr Male 10 10 8 3 Female 10 10 12 7 0.7 What is the probability that a random person is a woman or a freshman? From the table, 49/70 = 0.7 0.08 You draw three people at random. What is the probability that all three are males? ( 31 3 ) ( 39 0 ) ( 70 / 3 ) = 0.0821. 3. In 1930, the Health and Nutrition Examination took a random sample of adults and recorded their weight. In 2005, the CDC looked at the lifespans of those participants and found that people who weighed less lived longer. Based on this, the CDC decided that obesity was a critical health issue. Yes Was this an observational study? What confounding variable might explain these findings? Why? Age. People tend to gain weight as they grow older. So people with low weights tend to be young, and have more time on their clocks. 4. Suppose that 30% of the patrons in a bar are women. You measure everyone s height. 3
F True or False: You expect the median to be less than the average. The mean is pulled down by the smaller heights of the women, but the median is resistant to outliers. T True or False: You expect the sd to be larger than you would find in a bar whose clientele was exclusively male. Because there is more spread in the data when it is a mixture of shorter women and taller men. 5. Consider the following numbers: 6.5 (a) What is the IQR? 1, 8, 9, 10, 7, 8, 4, 2 As in class, the 25th percentile is any number between 1 and 2, thus 1.5, and the 75th percentile is between 8 and 8, or 8. And 8-1.5 = 6.5. 5.85 What is the standard deviation? Routine calculation. Note that there is no reason to think that this is a sample from a larger population. 30 Suppose a set of numbers has standard deviation equal to 6. If you multiply each number by -5 and subtract 10, then what is the new range? The new sd is the absolute value of the multiplier times the old, or 5 * 6. 0 Suppose a set of numbers has median equal to -2. If you multiply each number by -5 and subtract 10, then what is the new median? The new median is the same operation applied to the old, or (-5)*(-2) - 10 = 0. 4
0.68 to 0.7 6. A decorticator is a device for shelling nuts (usually by shooting the nut at a surface with a precise amount of force and orientation). To test your new decorticator, you fire 100 nuts. Suppose the decorticator correctly extracts the meat 90% of the time. What is the approximate probability that more than 88 nuts are properly shelled? Use the normal approximation to the binomial with a continuity correction. The mean of the binomial is np = 90 and the standard deviation is np(1 p) = 3. Thus the probability that X > 88 is approximately the probability that z > (88.5 90)/3. 7. The joint density of (X,Y ) is f(x,y) = 6(x y) for 0 < y < x < 1. What is the marginal density of y? Integrate the joint density with respect to x; note that the limits of integration for x are from y up to 1. You get f Y (y) = 3(1 2y + y 2 ) on 0 y 1. What is the conditional density of x given y? Since f X Y = f(x,y)/f Y (y), then the conditional density is f X Y (x y) = 2(x y)/(1 2y + y 2 ). 5/6 What is the expected value of X when y = 0.5? IE[X Y = 0.5] = 1 0.5 x2(x 0.5)/(1 2(0.5) + (0.5)2 )dx = 5/6. 1/80 What is the covariance between X and Y? (Assume that IE[X] = 3/4.) We need IE[XY ] and IE[Y ]. IE[XY ] = 1 x 0 0 xy 6(x y)dydx = 1/5 5
and IE[Y ] = 1 0 y 3(1 2y + y 2 )dy = 1/4 so the covariance is IE[XY ] IE[X] IE[Y ] = (1/5) (1/4)(3/4) = 1/80. 1/3 What is the correlation between X and Y? (Assume that Var [X] = 3/80.) The correlation is the covariance divided by the product of the standard deviations of X and Y. The standard deviation of X is given as 3/80. To find the variance of Y we first need IE[Y 2 ] = 1 0 y2 3(1 2y y 2 )dy = 1/10. Then Var [Y ] = 1/10 (1/4) 2 = 3/80. So the standard deviation of Y is 3/80. Thus the correlation is 1/3. Note: You could have immediately deduced that the variances of X and Y are equal, by the symmetry in the joint density function. 8. There are three brands of medical pacemakers. Heart Throb controls 40% of the market, The Body Electric controls 25%, and the rest is owned by The Beat Goes On. The failure rate for Heart Throb devices is 1%. The failure rate for The Body Electric is 2%. And the failure rate for The Beat Goes On is 4%. If the chief engineer at the FDA s Center for Devices and Radiological Health dies from a pacemaker failure, what is the probability that he carried a Heart Throb? Bayes rule. (0.01)*(0.4)/[(0.01)*(0.4) + (0.02)*(0.25) + (0.04)*(0.35)] = 0.1739. 9. Los Alamos controls 20 missiles, of which 8 are inoperable. In order to renew their contract from the Department of Energy, they have to successfully demonstrate two launches. They randomly choose a missile to launch, try it, and continue until they have had two successful launches. What is the probability that they terminate on the third try? (Comment: This is called inverse sampling, and is useful when each try is risky or expensive.) In order to get the second success on the third trial, there must have been one success and one failure on the first two trials, followed by a success. The probability of a success and failure on the first two is hypergeometric with 12 successes, 8 failures, 2 ). Given that outcome, the probability ( ) ( ) ( 8 12 20 draws, and 1 success, or / 1 1 2 that the next launch is a success is 11/18. The product of these is 0.3088. 6
120 10. Shakespeare wrote 10 historical plays. An English professor wants to pick three to assign to his class for reading. In how many ways could he select them? (Assume that the order does not matter.) ( 10 3 ) 60 11. In how many ways can 6 people sit at a circular table? (That is, one s neighbors matter but not which chair one selects and not whether your neighbor sits to the left or right.) Imagine that the chairs are numbered. Then there are 6! ways for the people to sit in order. But since this is a circle, 6 of those are equivalent. And since left/right doesn t matter, 2 more arrangements are equivalent. Thus there are 6!/(6*2) = 60 ways to do this. 0.53 12. On average, people have 2.8 car accidents. What is the chance that a random person will have three or more accidents in their life? Poisson. 1 2.8 0 /0!e 2.8 2.8 1 /1!e 2.8 2.8 2 /2!e 2.8 =.5305. 0.32 13. Suppose that 20% of students love statistics. What is the chance that three or more people among ten whom you meet at The Perk enjoy their statistics course? ( 10 Binomial. 1 0 ) ( 0.2 0 0.8 10 10 1 ) ( 0.2 1 0.8 9 10 2 ) 0.2 2 0.8 8 = 0.3222. 14. A Poisson process describes the timing of random events, such as celebrity deaths. The time between events is exponential with rate λ, and the number of events in a unit time period has the Poisson distribution with parameter λ. 1 exp( λ) What is the probability that time between two successive events is less than one day? 7
1 exp ( λ) What is the formula for the probability that the count is 1 or more in one day? Explain. In a Poisson process, if the waiting time is greater than one unit, then no events occur in that time period. Thus the probability that the time is less than one unit has to equal the probability at one or more events occur. 8