University of California, Berkeley, Statistics 20, Lecture 1 Michael Lugo, Fall 2010 Exam 2 November 3, 2010, 10:10 am - 11:00 am Name: Signature: Student ID: Section (circle one): 101 (Joyce Chen, TR 9) 102 (Moorea Brega, TR 10) 103 (Moorea Brega, TR 11) 104 (Joyce Chen, TR 11) This exam consists of seven pages: this cover page, five pages each containing one question, and a table of the normal distribution. You may use a calculator, and notes on one side of a standard 8.5-by-11-inch sheet of paper which you have written by hand, yourself. You must show all work other than basic arithmetic. The total number of points available is 100. DO NOT WRITE BELOW THIS LINE Question: 1a 1b 1c 2a 2b 3a 3b 4a 4b 4c 4d 5a 5b Points available: 8 8 4 10 10 12 8 5 5 5 5 10 10 Your score: 1
1. The number of runs scored by Major League Baseball teams in 2010 and the number of games won satisfy: average number of runs: 710 SD of runs: 75 average number of wins: 81 SD of wins: 11 r = 0.8 (a) [8 points] Find the equation of the regression line for predicting the number of wins from the number of runs. (b) [8 points] Find the equation of the regression line for predicting the number of runs from the number of wins. (c) [4 points] If you solve the equation from (a) for the number of runs, you don t get the answer from (b). Why not? Solution. (a) This line passes through (runs, wins) = (710, 81) and has slope r SD(wins)/SD(runs) = 0.8 11/75. So the equation is (wins 81) = Solving for the number of wins gives 0.8 11 (runs 710). 75 wins = 0.1173 runs 2.307. (b) This line passes through (wins, runs) = (81, 710) and has slope r SD(runs)/SD(wuns) = 0.8 75/11. So the equation is (runs 710) = Solving for the number of runs gives 0.8 75 (runs 81). 11 runs = 5.45 wins + 268. (c) If solving the equation in (a) for the number of runs gave the equation in (b), then that would mean that the two regression lines were the same. But they are not. For example, assume that the scatter plot is football-shaped and sketch an ellipse to represent it. If we plot runs on the horizontal axis and wins on the vertical axis, then the line from (a) passes through the leftmost and rightmost points on this ellipse, while the line from (b) passes through the topmost and bottommost; you can see from such a picture that these aren t the same line. 2
2(a). [10 points] Compute exactly the probability that, in flipping a fair coin nine times, you get exactly six heads. 2(b). [10 points] Estimate the same probability using the normal distribution. Your answer to each part should be a single decimal or fraction. If you give your answer as a fraction, it does not need to be in lowest terms. So if the answer were 1/3, then 2/6 or.333 is acceptable, but (2/5) (5/6) is not. Solution. (a) From the binomial distribution, this is ( 9 6) /2 9. Computing, ( ) 9 = 9 8 7 6 3 2 1 = 84 and so the probability is 84/512 = 21/128 = 0.164. (b) The number of heads has average 9/2 and standard deviation 9 1/2 1/2 = 3/2. We want to compute then the probability that the normal distribution with average 4.5 and SD 1.5 is between 5.5 and 6.5; that s the area under the normal curve between (5.5 4.5)/1.5 = 0.67 and (6.5 4.5)/1.5 = 1.33. From the normal table this is about (1/2)(0.82 0.48) = 0.17. 3
3. In a certain far-off country, there are two sorts of people: Smurfs and Muppets. Smurfs are blue, while Muppets come in all colors. (These colors include blue examples of blue Muppets are Grover and Cookie Monster.) The traditional Smurf neighborhood is le Pays maudit, and the traditional Muppet neighborhood is Sesame Street. Due to much recent immigration, much of it undocumented, it is not known whether there are more Smurfs or more Muppets in the country. The authorities in this country wish to determine what proportion of the people living in their country are blue. They propose the following four sampling methods. ( ) Choosing 50 people at random on Sesame Street and 50 people at random in le Pays maudit and observing their color. ( ) Observing the color of 100 people passing by on Sesame Street. ( ) Choosing 100 people at random from the country as a whole and observing their color. ( ) Observing the color of 50 people passing by on Sesame Street, and 50 people passing by in le Pays maudit. (a) [12 points] Write 1 next to the symbol of the MOST accurate method, 2 next to the symbol of the second most accurate method, 3 next to the symbol of the third most accurate method, and 4 next to the symbol of the LEAST accurate method. Solution. : 2 : 4 : 1 : 3 (b) [8 points] Explain your reasoning in part (a). Solution. The method in is a simple random sample and is therefore the most accurate. All other things being equal, choosing at random will be more accurate than observing, which gives the investigator some latitude, so is better than. Finally, is better than. The method samples in both Sesame Street and le Pays maudit, although perhaps not with the right proportions;, on the other hand, ignores a very large part of the country entirely! 4
4. A box contains five tickets, two marked with stars, and the other three blank. Two draws are made at random without replacement from this box. (a) [5 points] What is the chance of getting a blank ticket on the first draw? (b) [5 points] What is the chance of getting a blank ticket on the second draw? (c) [5 points] What is the chance of getting a blank ticket on the second draw, given that you got a blank ticket on the first draw? (d) [5 points] What is the chance of getting at least one ticket marked with a star in the two draws? Your answer to each part should be a single decimal or fraction. If you give your answer as a fraction, it does not need to be in lowest terms. So if the answer were 1/3, then 2/6 or.333 is acceptable, but (2/5) (5/6) is not. Solution. (a) 3/5, since there are three blank tickets out of five. (b) 3/5. Same as (a), by symmetry. Or note that the probability of getting a blank followed by a blank is (3/5)(2/4), and the probability of getting a star followed by a blank is (2/5)(3/4); these add up to 3/5. (c) 2/4, since there are two blank tickets left out of four once we draw the first one. (d) This is one minus the probability of getting two blank tickets, which is (3/5)(2/4) = 6/20 = 3/10; so the answer is 7/10. 5
5. Consider a die with the numbers 1, 2, 2, 4, 5, 10 on its sides. (a) [10 points] What is the probability of getting a sum of 6 if we roll this die twice? (b) [10 points] Estimate the probability of getting below 370 as the sum of one hundred rolls of this die. Solution. (a) Denote the 2s as 2 A and 2 B. Then we can roll a sum of 6 as 1 + 5, 2 A + 4, 2 B + 4, 4 + 2 A, 4 + 2 B, 5 + 1 there are six ways to roll a 6, out of a total of 6 6 = 36. The probability is 6/36 = 1/6. (b) The average result from a single roll of this die is (1 + 2 + 2 + 4 + 5 + 10)/6 = 4 and the standard deviation is (4 1)2 + (4 2) 2 + (4 2) 2 + (4 4) 2 + (4 5) 2 + (4 10) 2 6 = 54 6 = 3. So the average result from 100 rolls is 4 100 = 400, with SD 3 100 = 30. So 370 is one SD less than the average; the probability is about 16 percent. 6
A NORMAL TABLE Height gives the height of the normal curve at z, in percent. Area gives the area under the normal curve between z and z, in percent. z Height Area z Height Area z Height Area 0.00 39.89 0 1.50 12.95 8.64 3.00 0.443 99.730 0.05 39.84 3.99 1.55 12.00 87.89 3.05 0.381 99.771 0.10 39.70 7.97 1.60 11.09 89.04 3.10 0.327 99.806 0.15 39.45 11.92 1.65 10.23 90.11 3.15 0.279 99.837 0.20 39.10 15.85 1.70 9.40 91.09 3.20 0.238 99.863 0.25 38.67 19.74 1.75 8.63 91.99 3.25 0.203 99.885 0.30 38.14 23.58 1.80 7.90 92.81 3.30 0.172 99.903 0.35 37.52 27.37 1.85 7.21 93.57 3.35 0.146 99.919 0.40 36.83 31.08 1.90 6.56 94.26 3.40 0.123 99.933 0.45 36.05 34.73 1.95 5.96 94.88 3.45 0.104 99.944 0.50 35.21 38.29 2.00 5.40 95.45 3.50 0.087 99.953 0.55 34.29 41.77 2.05 4.88 95.96 3.55 0.073 99.961 0.60 33.32 45.15 2.10 4.40 96.43 3.60 0.061 99.968 0.65 32.30 48.43 2.15 3.96 96.84 3.65 0.051 99.974 0.70 31.23 51.61 2.20 3.55 97.22 3.70 0.042 99.978 0.75 30.11 54.67 2.25 3.17 97.56 3.75 0.035 99.982 0.80 28.97 57.63 2.30 2.83 97.86 3.80 0.029 99.986 0.85 27.80 60.47 2.35 2.52 98.12 3.85 0.024 99.988 0.90 26.61 63.19 2.40 2.24 98.36 3.90 0.020 99.990 0.95 25.41 65.79 2.45 1.98 98.57 3.95 0.016 99.992 1.00 24.20 68.27 2.50 1.75 98.76 4.00 0.013 99.994 1.05 22.99 70.63 2.55 1.54 98.92 4.05 0.011 99.995 1.10 21.79 72.87 2.60 1.36 99.07 4.10 0.009 99.996 1.15 20.59 74.99 2.65 1.19 99.20 4.15 0.007 99.997 1.20 19.42 76.99 2.70 1.04 99.31 4.20 0.006 99.997 1.25 18.26 78.87 2.75 0.91 99.40 4.25 0.005 99.998 1.30 17.14 80.64 2.80 0.79 99.49 4.30 0.004 99.998 1.35 16.04 82.30 2.85 0.69 99.56 4.35 0.003 99.999 1.40 14.97 83.85 2.90 0.60 99.63 4.40 0.002 99.999 1.45 13.94 85.29 2.95 0.51 99.68 4.45 0.002 99.999 7