Solutions to Homework 1

1 Solutions to Homework 1 Statistics 302 Professor LargetTextbook Rock-Paper-ScissorsRock-Paper-Scissors, also called Roshambo, is a popular two-playergame often used to quickly determine a winner and loser. In the game, each player puts out a fist(rock), a flat hand (paper), or a hand with two fingers extended (scissors). In the game, rock beatsscissors which beats paper which beats rock. The question is: Are the three options selected equallyoften by players? Knowing the relative frequencies with which the options are selected would givea player a significant advantage. A study observed 119 people playing Rock-Paper-Scissors. Theirchoices are shown the table.

2 (a) What is the sample in this case? What is the population? What does the variable measure?(b) Construct a relative frequency table of the results.(c) If we assume that the sample relative frequencies from part (b) are similar for the entirepopulation, which option should you play if you want the odds in your favor?(d) The same study determined that, in repeated plays, a player is more likely to repeat theoption just picked than to switch to a different option. If your opponent just played paper,which option should you pick for the next round?Option Selected FrequencyRock66 Paper39 Scissors14 Total119 Solution(a) The sample is the 119 players who were observed.

3 The population is all people who playrock-paper-scissors. The variable records which of the three options each player plays. This is acategorical variable.(b) A relative frequency table is shown below. We see that rock is selected much more frequentlythan the others, and then paper, with scissors selected least Selected Relative (c) Since rock is selected most often, your best bet is to play paper.(d) Your opponent is likely to play paper again, so you should play Home Field Advantage in SoccerIn the bookScorecasting, we learn that Across 43professional soccer leagues in 24 different countries spanning Europe, South America, Asia, Africa,Australia, and the United States (covering more than 66,000 games), the home field advantage[percent of games won by the home team] in soccer worldwide is Is this a population or asample?

4 What are the cases and approximately how many are there? What is the variable and is1it categorical or quantitative? What is the relevant statistics, including correct notation?SolutionThe dataset includes all professional soccer games, so this is a population. The cases are the soccergames, and there are approximately 66,000. The variable is whether or not the home team wonthe game, and it is categorical. The relevant statistic isp= Smoking and Pregnancy RateStudies have concluded that smoking while pregnant canhave negative consequences, but could smoking also negatively one s ability to become pregnant?A study collected data on 678 women who had gone off birth control with the intention of becomingpregnant.

5 Smokers were defined as those who smoked at least one cigarette a day prior to preg-nancy. We are interested in the pregnancy rate during the first cycle off birth control. The resultsare summarized in the table Non-SmokerTotalPregnant38206244 Not Pregnant97337424 Total135543678(a) Is this an experiment or an observational study? Can we use these data to determine whethersmoking influences one s ability to get pregnant? Why or why not?(b) What is the population of interest?(c) What is the proportion of women successfully pregnant after their first cycle ( p)? Proportionof smokers successful ( ps)? Proportion of nonsmokers successful ( pns)?

6 (d) Find and interpret ( pns ps) the difference in proportion of success between non-smokers (a) Since no one assigned smoking or not to the participants, this is an observational study. Becausethis is an observational study, we can not use this data to determine whether smoking influencesone s ability to get pregnant. We can only determine whether there is an association betweensmoking and ability to get pregnant.(b) The sample collected is on women who went off birth control in order to become pregnant,so the population of interest is women who have gone off birth control in an attempt to becomepregnant.

7 (c) We look in the total section of our two way table to find that out of the 678 women at-tempting to become pregnant, 244 succeeded in their first cycle, so p= 244/678 = Forsmokers we look only in the Smoker column of the two way table and observe 38 of 135 succeeded,so ps= 38/135 = For non-smokers we look only in the Non-smoker column of the two waytable and observe 206 of 543 succeeded, so pns= 206/543 = (d) For the difference in proportions, we have pns ps= = This means thatin this sample, the percent of non-smoking women successfully getting pregnant in the first cycleis 10 percentage points higher than the percent of Which of These Things Is Not Like the Other?

8 Four students were working togetheron a project and one of the parts involved making a graph to display the relationship in a two-waytable of data with two categorical variables: college accept/reject decision and type of high school(public, private, parochial). The graphs submitted by each student are shown in the book. Threeare from the same data, but one is inconsistent with the other three. Which is the bogus graph? (b) is the impostor. It shows more parochial students than private school students. Theother three graphs have more private school students than Fiber in the DietThe number of grams of fiber eaten in one day for a sample of ten people are10 11 11 14 15 17 21 24 28 115(a) Find the mean and the median for these data.

9 (b) The value of 115 appears to be an obvious outlier. Compute the mean and the median for thenine numbers with the outlier excluded.(c) Comment on the effect of the outlier on the mean and the (a) The mean is x=10+11+11+14+15+17+21+24+28+11510=26610 = median is15+172= 16.(b) Without the outlier, we have x= Since n = 9, the median is the middle have m = 15.(c) The outlier has a very significant effect on the mean and very little effect on the Beta-Carotene Levels in the BloodThe plasma beta-carotene level (concentration ofbeta-carotene in the blood), in ng/ml, was measured for a sample ofn= 315 individuals, and theresults are shown in the histogram in the book.

10 (a) Describe the shape of the distribution. Is it symmetric or skewed? Are there any obviousoutliers?(b) Estimate the median of this sample.(c) Estimate the mean of this (a) The distribution has a right skew. There are a number of apparent outliers on the right side.(b) The actual median is 140 ng/ml. Estimates between 120 and 160 are reasonable.(c)The actual mean is ng/ml. Estimates between 160 and 220 are reasonable. Note thatthe outliers and right skew should make the mean larger than the Does Sexual Frustration Increase the Desire for Alcohol?Apparently, sexual frus-tration increases the desire for alcohol, at least in fruit flies.