Econ 57 Fall 2001 Midterm Answers
1. The probability of five heads is(1/2)5 = 1/32; the probability of five tails is (1/2)5 = 1/32. Therefore, the probability of five heads or five tails is 2/32 = 0.0625.
2. Using Bayes Rule,
3. All three cases can be answered by seeing which city has the SMALLER standardized Z value:
a. ZA = (100 - 80)/20 = 1, ZB = (100 - 80)/30 = 2/3. So, B.
b. ZA = (100 - 80)/20 = 1, ZB = (100 - 70)/20 = 3/2. So, A.
c. ZA = (100 - 80)/20 = 1, ZB = (100 - 70)/30 = 1. So, equal.
4. The total area of the bars is not 1.
5. Because Mark has a higher probability of snow, he should bet on snow and Mindy should bet against snow. So, we will make the bet that Mindy pays Mark $X if it snows and Mark pays Mindy $Y if it doesnt.
From Marks viewpoint, his expected value is (X)(4/5) + (-Y)(1/5). For this to be positive, he needs X/Y > 1/4. From Mindys viewpoint, her expected value is (-X)(2/3) + (Y)(1/3). For this to be positive, she needs X/Y < 1/2. Thus, any bet with 1/4 < X/Y < 1/2 will give both persons positive expected values. For example, Mindy pays Mark $1 if it snows and Mark pays Mindy $3 if it doesnt.
6. Here are the answers:
a. scatter diagram
b. scatter diagram
c. side-by-side boxplots
d. time series graph
e. scatter diagram
7. Marilyns correct answer: Id choose heads/tails (HT), because Id win three out of four times! Thats because there are four different sequence combinations: HH, HT, TH, and TT. If tails/tails (TT) were to appear at the very start, youd win, but that would happen only one-fourth of the time. For TT to appear any time afterward, it would have to be preceded by H. Which means that Id win before you ever saw your sequence come up at all!
8. Using the binomial distribution with p = 0.2 and n = 5:
a. P[X = 5] = 0.25.
9. The mean is 2.3, the median is 2, the standard deviation is 1.
10. Marilyns (correct) answer [Ask Marilyn, Parade, October 29, 2000.]:
The distribution of sexes will remain roughly equal. Thats becauseno matter how many or how few children are born anywhere, anytime, with or without restrictionhalf will be boys and half will be girls! Only the act of conception (not the government!) determines the sex.
One can demonstrate this mathematically. (In this case, well assume that women with firstborn girls will always have a second child.) Lets say 100 women give birth, half to boys and half to girls, The half with boys must end their families. There are now 50 boys and 50 girls. The half with girls (50) give birth again, half to boys and half to girls. This adds 25 boys and 25 girls, so there are now 75 boys and 75 girls. Now all must end their families. So the result of the policy is that there will be fewer children in number, but the boy/girl ratio will not be affected.