Download Quantitative Methods Midterm: Probability & Hypothesis Testing Solutions and more Exams Introduction to Public Administration in PDF only on Docsity! PUAF 610 QUANTITATIVE METHODS Fall 1995 MIDTERM EXAM SOLUTIONS 1. The World Series is a best-of-seven series between two teams (i.e., the first team to win four games wins the Series). A. Assuming that the teams that play in the World Series are evenly matched (i.e., probability of winning a game = 0.5 for both teams), what is the probability that a team would win the Series after having a 3–1 lead? (10 points) There is only one way for the team that is ahead to lose the series: to lose each of the next three games. If the probability of winning each game is independent: Pwin series = 1 – Plose series = 1 – (Plose game) 3 = 1 – 0.53 = 1 – 0.125 = 0.875 = 87.5% B. The following statement appeared in Thursday’s Washington Post: “Of the previous 35 teams to have 3–1 World Series leads, 29 went on to capture the Series.” Does this sample correspond to what you would have expected from part A? Test the null hypothesis that any deviation is due to sampling error. (15 points) H0: π = 0.875 HA: π ≠ 0.875 p = 29/35 = 0.8286 ( ) ( )π − π σ = = =p 1 0.875 0.125 0.0559 n 35 − π − − = = = = − σp p 0.8286 0.875 0.0464 z 0.83 0.0559 0.0559 I would conclude that experience is consistent with what one would expect based on simple probability theory; in statistical jargon, I would reject the null hypothesis (p = 0.4 two-tailed). Note that n(1–π) = 35(0.125) = 4.4, so the normal approximation is not good, but it doesn’t matter in this case because z is so small. 2. Does alcohol consumption during pregnancy adversely affect the fetus? One study found that 58 women who averaged two or more drinks per day had significantly smaller babies than 104 who did not drink at all during their pregnancies: an average of 103 ounces for drinkers versus 121 ounces for nondrinkers. (The standard deviation deviations were 35 and 25 ounces, respectively.) A. Do these results constitute convincing evidence that alcohol consumption lowers birth weights? Formulate null and alternative hypotheses, and test the null hypo- thesis. How likely is it that the observed difference is due to chance? (20 points) H0: µdrinker = µnon-drinker HA: µdrinker ≠ µnon-drinker (µdrinker < µnon-drinker also acceptable) − − − −= ≅ = = = σ + +1 2 1 2 1 2 2 2 2 2 x x 1 2 1 2 x x x x 121 103 18 z 3.46 5.2s s 25 35 n n 104 58 Reject the null hypothesis (p = 0.0003 one-tail, 0.0006 two-tail); this study provides evidence that drinkers give birth to lower birth-weight babies. B. What is wrong with the design of this study? (Hint: think about how participants probably were selected.) (5 points) It is highly unlikely that participants were randomly selected and randomly divided into drinking and non-drinking groups. Participants in each group were self- selected. This is a fatal design flaw, because drinkers are also more likely to smoke, eat poorly, receive poor prenatal care, etc. Unless these other factors are somehow controlled or accounted for, the study cannot show the effect of drinking on birthweight.
