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STAT 31 Practice Midterm 2 Fall, 2005 INSTRUCTIONS: BOTH THE BUBBLE SHEET AND THE EXAM WILL BE COLLECTED. YOU MUST PRINT YOUR NAME AND SIGN THE HONOR PLEDGE ON THE BUBBLE SHEET. YOU MUST BUBBLE-IN YOUR NAME & YOUR STUDENT IDENTIFICATION NUMBER. YOU MUST PUT YOUR NAME ON THE EXAM ALSO. EACH QUESTION HAS ONLY ONE CORRECT CHOICE (decimals may need rounding). USE "NUMBER 2" PENCIL ONLY - DO NOT USE INK - FILL BUBBLE COMPLETELY. NO NOTES OR REMARKS ARE ACCEPTED - DO NOT TEAR OR FOLD THE BUBBLE SHEET. A GRADE OF ZERO WILL BE ASSIGNED FOR THE ENTIRE EXAM IF THE BUBBLE SHEET IS NOT FILLED OUT ACCORDING TO THE ABOVE INSTRUCTIONS. REAL EXAM WILL INCLUDE 25 QUESTIONS. EACH QUESTION IS WORTH 1 POINT. 1. The least-squares regression line is A) the line that makes the square of the correlation in the data as large as possible. B) the line that makes the sum of the squares of the vertical distances of the data points from the line as small as possible. C) the line that best splits the data in half, with half of the points above the line and half below the line. D) all of the above. 2. Let the random variable X represent the profit made on a randomly selected day by a certain store. Assume X is normal with a mean of $360 and standard deviation $50. The value of P(X > $400) is A) 0.2881 B) 0.8450 C) 0.7881 D) 0.2119 3. If A and B are mutually exclusive events with P(A) = .70, then P(B): A) can be any value between 0 and 1 B) can be any value between 0 and .70 C) cannot be larger than .30 D) cannot be determined with the information given Page 1 4. Event A occurs with probability 0.3 and event B occurs with probability 0.4. If A and B are independent, we may conclude A) P(A and B) = 0.12 B) P(A|B) = 0.3 C) P(B|A) = 0.4 D) all of the above. 5. A researcher wishes to study how the average weight Y (in kilograms) of children changes during the first year of life. He plots these averages versus the age X (in months) and decides to fit a least-squares regression line to the data with X as the explanatory variable and Y as the response variable. He computes the following quantities. r = correlation between X and Y = 0.9 X = mean of the values of X = 6.5 Y = mean of the values of Y = 6.6 s X = standard deviation of the values of X = 3.6 sY = standard deviation of the values of Y = 1.2 The intercept of the least-squares line is A) 4.52 B) -12.9 C) 4.65 D) -13.3 6. Refer to the problem above (problem 5). The slope of the least-squares line is A) 0.30 B) 0.88 C) 1.01 D) 3.0 Page 2 7. Refer to Problem 5 for the background. Suppose using the same data, the researcher wishes to predict the children age from their average weight. That means, he needs to fit a least-squares regression line to the data with Y (average weight) as the explanatory variable and X (age) as the response variable. The slope of the least-squares line is A) 3.33 B) 0.30 C) 2.70 D) 4.65 8. If X and Y are random variables with σ X2 = 7.5 , σ Y2 = 6 and ρ ( X , Y ) = 0.6 , then σ 22X +3Y is approximately: A) 33 B) 37 C) 88 D) 132 9. Ignoring twins and other multiple births, assume babies born at a hospital are independent events with the probability that a baby is a boy and the probability that a baby is a girl both equal to 0.5. The probability that the next three babies are of the same sex is A) 1.0 B) 0.125 C) 0.250 D) 0.500 10. Suppose that you are a student worker in the library and they agree to pay you using the Random Pay system. Each week your supervisor flips a fair coin. If it comes up head, your pay for the week is $80 and if it comes up tail your pay for the week is $40. What is the expected value of your weekly pay? A) $20 B) $40 C) $60 D) $80 Page 3 11. Refer to the problem above (Problem 8). What is the standard deviation of your weekly pay? A) 20 B) 28 C) 400 D) 2 12. Event A occurs with probability 0.4. The conditional probability that A occurs given that B occurs is 0.5, while the conditional probability that A occurs given that B does not occur is 0.2. What is the conditional probability that B occurs given that A occurs? A) 0. B) 4/7. C) 5/6. D) None of the previous is right or cannot be determined from the information given. 13. Which of the following statements is correct given that the events A and B have nonzero probabilities? A) A and B cannot be both independent and mutually exclusive B) A and B can be both independent and mutually exclusive C) A and B are always independent D) A and B are always mutually exclusive 14. The weight of medium-size tomatoes selected at random from a bin at the local supermarket is a random variable with mean µ = 10 oz. and standard deviation σ = 1 oz. Suppose we pick two tomatoes at random from the bin. The difference in the weights of the two tomatoes selected (the weight of first tomato minus the weight of the second tomato) is a random variable with a standard deviation (in ounces) of A) 0.00 B) 1.00 C) 1.41 D) 2.00 Page 4 15. Suppose X is a continuous random variable taking values between 0 and 2 and having the probability density function below. 1 0 0 1 2 P(1 ≤ X ≤ 2) has value A) 0.50 B) 0.33 C) 0.25 D) 0.00 16. According to the 1990 census, those states that had an above average number X of people who failed to complete high school tended to have an above-average number Y of infant deaths. In other words, there was a positive association between X and Y. The most plausible explanation for this association is A) X causes Y. Thus programs to keep teens in school will help reduce the number of infant deaths. B) Y causes X. Thus programs that reduce infant deaths will ultimately reduce the number of high school drop-outs. C) changes in X and Y are due to a common response to other variables. For example, states with large populations will have larger numbers of people who fail to complete high school and a larger number of infant deaths. D) the association between X and Y is purely coincidental. It is implausible to believe the observed association could be anything other than accidental. 17. We roll a pair of standard fair dice and observe the total number of dots on the top faces. Our sample space is S={2,3,4,5,6,7,8,9,10,11,12}. What's the probability that the total number of dots is at least 4? A) 5/6 B) 11/12 C) 9/11 D) 8/9 Page 5 18. Three prisoners, A, B, and C are on death row. The governor decides to pardon one of the three and chooses at random the prisoner to pardon. He informs the warden of his choice but requests that the name be kept secret for a few days. The next day, A tries to get the warden to tell him who had been pardoned. The warden refuses. A then asks which of B or C will be executed. The warden thinks for a while, then tells A that B is to be executed. If A were pardoned, the warden could tell A that either B or C would die. Suppose the probability for the warden to tell A that B dies is p. (p=1/2 is the case we talked about in class where the warden is impartial between B and C.) As a result, the probability for A to be pardoned if the warden tells him that B dies can be expressed as a function of p. Which function should it be? A) 1/(p+1) B) 1/3 C) p/(1+p) D) 1/(2+2p) Page 6 Answer Key 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. B D C D C A C D C C A C A C C C B C Page 7