Model Project Number 1: Trivial Pursuit of Statistics
Jessica Cottreau Picture this, you and a few people are sitting around and decide to play a friendly game of Trivial Pursuit. Everything is going well, everyone is having a good time. Then you are faced with a question for a piece of pie, Who placed the rst telephone call to the moon? Your friend looks at the back of the card, gazes at the answer, and annoyingly says, Thats easy. You really do not know the answer, so you choose an answer from the sea of names swimming in your head. Uunfortunately it was not the right answer and your friend with the card does not attempt to conceal your momentary idiocy, instead yells out Richard Nixon in disbelief. But did your friend really know the answer? It is possible that your friend truly knew the answer to the question before he or she saw the answer, but it is also possible that your friend fell victim to the I knew it all along effect. The= I knew it all along effect is a psychologically researched effect that states that a person tends to overestimate his or her knowledge on a subject when the answers are presented. Spurred by my observations of this effect while playing Trivial Pursuit, I decided to test these observations statistically. I began my study by handpicking ve questions from Parker Brothers Trivial Pursuit Genus IV board game. One question was selected from the People & Places, Arts & Entertainment, History, Science & Nature, and Sports & Leisure categories. The Wild Card category was left out to avoid two questions on the same topic. Two question sheets were then designed, one with the ve questions, and the other with the questions followed by the answers. After each question set the participant is asked whether the questions were easy or difcult. Treatment 1no answers Answer the following questions in your head, you do not need to write the answers down. 1. Who directed Vertigo in 1958? 2. What African republics name was inspired by its thriving elephant tusk trade? 3. Who placed the rst telephone call to the moon? 4. What isnt found in liquid form on any other planet but earth? 5. Who was involved as a player or coach in three super bowls with the Cowboys, two with the Eagles and one with the Bears? After answering these questions did you nd them easy or difcult? Put a check next to one of the following statements. They were easy-had difculty with 0-2 of the questions. They were difcult-had difculty with 3-5 of the questions. Treatment 2answers Answer the following questions in your head, the answers are recorded after each question.
1. Who directed Vertigo in 1958? Alfred Hitchcock. 2. What African republics name was inspired by its thriving elephant tusk trade? Ivory Coast. 3. Who placed the rst telephone call to the moon? Richard Nixon. 4. What isnt found in liquid form on any other planet but earth? Water. 5. Who was involved as a player or coach in three super bowls with the Cowboys, two with the Eagles and one with the Bears? Mike Ditka. After answering these questions did you nd them easy or difcult? Put a check next to one of the following statements. They were easy-had difculty with 0-2 of the questions. They were difcult-had difculty with 3-5 of the questions. The two treatments were then ordered randomly by putting numbers 1-100 in a bucket and picking out fty of the numbers for the rst treatment and then putting the two treatments in the random order they were assigned. Data The alternative chosen for this experiment is the second alternative, where there will begreater successes on the treatment that provided the subject with answers. Treatment 1 (No answers) 2 (Answers) Total Easy 35 Difcult 65 Total 100 Row Proportions Easy Difcult Total 0.24 0.76 1.00 0.46 0.54 1.00

Thus, x = p1 p2 = 0.24 0.46 = 0.22. (Jessica chose to use the continuity correction in order to obtain a more accurate approximation.) New x = 0.22 + 0.02 = 0.20, and = [35(65)]/[50(50)(99)] = 0.0959. Thus, z = 0.20/0.0959 = 2.09, and z = 2.09. The approximate P-value is 0.0183. From this P-value, you can determine that the experiment and results are statisticallysignicant. In conclusion, this study shows that the I knew it all along effect has some validity in the world today. It shows that people are likely to overestimate their ability to answer a question when the answer is in front of them. It also gives me the hope that my friend really didnt know that Richard Nixon was the rst person to place a telephone call to the moon.
Model Project Number 2: Sexual Harassment Study
Erica Olsen I performed a completely randomized design on fty female students between the ages of eighteen and twenty-two on the University of Wisconsin, Madison campus. The experiment had a dichotomous response to investigate the inuence of gender on sexual harassment. The two versions of the question that was asked are as follows. Version 1: A 45 year-old man is interviewing a 25 year-old woman for a political position in the state capital in Madison, Wisconsin. The interview is conducted at a local restaurant. Upon the arrival of the potential employee, the interviewer insists on buying her a drink. The interviewer asked three principal questions. 1. Are you currently in or looking for a stable relationship? 2. How many people have you been romantically involved with in the past year? 3. Do you plan to have children in the future? Is this interview a form of sexual harassment? YES NO
Version 2: A 45 year-old woman is interviewing a 25 year-old man for a political position in the state capital in Madison, Wisconsin. The interview is conducted at a local restaurant. Upon the arrival of the potential employee, the interviewer insists on buying him a drink. The interviewer asked three principal questions. 1. Are you currently in or looking for a stable relationship? 2. How many people have you been romantically involved with in the past year? 3. Do you plan to have children in the future? Is this interview a form of sexual harassment? YES NO
I randomized the treatments the subjects were assigned by using a stack of fty well-shufed cards, lying face down. Twenty-ve of the cards were marked with a 1, and twenty-ve of the cards were marked with a 2. Subjects took the top card from the stack. Subjects drawing a 1 read version one, while subjects drawing a 2 read version two. Subjects placed their cards in a box after answering either yes or no. During data collection, I was pleasantly surprised to nd that subjects were very willing to participate in my study. The collection process went very smoothly, and the randomization turned out to be less complicated than I had originally imagined. I am a member of the Kappa Alpha Theta sorority at the University of Wisconsin, Madison. I attended one of our weekly chapter meetings and set up the randomization technique mentioned above in order to assign treatments. These subjects are of interest to me because I t into the same gender category, age range, and am a student at the same University as the subjects. I felt that accumulating data from subjects with similar interests and experiences as myself would generate interesting results. The issue of sexual harassment is of interest to me because it has become a big issue in todays world. There are a good deal of 3

misconceptions as to what constitutes sexual harassment and who can be a victim of it. I wanted to study the changes in results based on which gender is experiencing sexual harassment and which gender is committing harassment. The theme of sexual harassment is especially pertinent to students on college campuses, so I utilized my student status to get feedback from other students. Before beginning data collection, I performed Fishers test for the appropriate alternative. I assumed the skeptic was correct, and therefore that the null hypothesis H0 : p1 = p2 was also correct. I chose an alternative hypothesis, H1 : p1 > p2 , or that there would be more successes on treatment one than on treatment two. I chose this hypothesis because of the way I feel the majority of society sees sexual harassment, as the victimization of a woman by a man, and not the other way around. I felt it was inconceivable that more people would feel that the woman was harassing the man, than would feel that the man was harassing the woman. After collecting the data, I found that there were indeed more successes on treatment one than treatment two. Treatment one had 23 successes out of 25 possible answers while treatment two had 15 successes out of 25. The observed value of the test statistic is x = 0.32. In addition, s = 0.122, and z = 2.62. Using these data and the standard normal curve, I obtained an approximate P-value of 0.0044. The data are presented in the table below. Treatment 1 (Male interviewer) 2 (Female interviewer)) Total Harassment? Yes No Total 50 Row Proportions Yes No Total 0.92 0.08 1.00 0.60 0.40 1.00
The data collected are highly statistically signicant according to the interpretation of the P-value. This means that the data support the alternative. While these results tell a lot about my subjects, they are not representative of an entire population or even the student population at UWMadison because my subjects were all female and within a relatively small age range. I have learned, within my subject group, that more subjects feel that sexual harassment occurred when the woman was interviewed by the man than when the man was interviewed by the woman. This correlates with my alternative hypothesis and shows that gender is a factor in drawing the line between sexual harassment and friendly conversation.

Model Project Number 3: Double Shot Basketball Lisa Mattiacci
In my house, my roommates and I have an electronic basketball shooting game called Double Shot. We love playing Double Shot and we are constantly competing against one another to gain the high score. I performed a balanced CRD to determine the effect that the use of different basketballs have on my scores. I shot for 30 seconds (the time allotted in the game). In the rst 20 seconds, each basket is worth 2 points. In the last 10 seconds, each basket is worth 3 points. The response was the total number of points I accumulated in the 30 seconds. The rst treatment was shooting with the ofcial Double Shot basketball. The second treatment was shooting with a Milwaukee Bucks basketball that was slightly heavier and at. I performed 40 trials, randomly drawing from a shufed stack of 40 cards, 20 red and 20 black. (The red cards represented Treatment 1.) Before collecting the data, I selected the rst alternative, 1 > 2 ; that is, I believed that my scores would be higher on Treatment 1 than on Treatment 2. Below are the sorted scores with the ofcial Double Shot basketball (Treatment 1): 27 36
Below are the sorted scores with the Milwaukee Bucks basketball (Treatment 2): 24 31
Below are stem-plots of the data, by treatment. Double Shot 6 Milwaukee Bucks 3
I constructed stem-plots because they display the exact data values and they also provide a clearly dened picture since the values can be grouped. I chose to sort the data into intervals of two because it showed a good amount of detail. The stem-plot for Treatment 1 does not reveal a prominent peak; instead it shows that the majority of the data are between 24 and 33 points. The stem-plot for Treatment 2 reveals a single peak and is relatively symmetric about that peak. These pictures agree with the summary statistics below.
Feature Measures of Center Mean Median Measures of Spread Range IQR Standard deviation
Double Shot 28 27.6.5 (= 31.5 25) 3.91
Milwaukee Bucks 14 4.5 (= 26 21.5) 3.61
After collecting the data, I found that the mean on Treatment 1 is 28 points and the mean on Treatment 2 is 24 points. The difference in means is 4 points. These data support my conjecture and are good proof because it is an average of 2 more baskets made which is relatively a lot considering the allotted time. If you also look at the quartiles of each treatment, in addition to the means, you will see even greater support for my conjecture. Quartile 1 signies that 25% of the data is below that point; Quartile 3 signies that 75% of the data is below that point. In Treatment 1, Q1= 25 and Q3 = 31.5. In Treatment 2, Q1 = 21.5 and Q3 = 26. Comparing the quartiles to the means of each treatment shows that at least 75% of the scores in Treatment 1 are larger than the average score of Treatment 2. This support can also be shown in terms of the IQR, which represents 50% of the data. The majority of the scores represented in the IQR for Treatment 1 are higher than the scores represented in the IQR for Treatment 2. The Range in each treatment is approximately equal which reveals that my accuracy was consistent in each treatment. There is a signicant difference in the maximum and minimum points I scored in each treatment though (Treatment 1 having higher endpoints). This difference is shown in the stem and leaf plots and also proves my alternative hypothesis. Finally, the standard deviation of this data is 3.91 for Treatment 1 and 3.61 for Treatment 2. The standard deviation of each treatment supports the empirical rule that 68% of the data lies within one standard deviation of the center (the sample mean of each treatment). This works well because the distributions of the basketball scores are bell-shaped, as shown by the stem-plots.

Model Project Number 4: Free Throws
Eric P. Seiler and Gesina M. Seiler We performed 100 dichotomous trials on free throw shooting. Ena was our shooter since she was a high school ( star ) shooting guard. She hasnt played in 13 years and we wondered if she could still shoot 82% like she used to. This project requires the use (or attempted use) of Bernoulli trials. We considered shooting the free throws to be Bernoulli trials and so proceeded to the basketball court. The following is a record of the shots taken. First 100 Trials: SSSFS SFSFF FFSSF SSSSS FFFSF FFSFF FSFSF FSSFS SSSFS SSSFS SFFFS SSFSF SFFFS FSFSF FSSSF SSSSF FSSFS SFFFS SFFFF SSSSF Note During the performance of our study, we thought we detected a pattern where it is obvious that there is no such thing. The following three assumptions must be true in order for a study to be Bernoulli trials. The possibility of only two outcomes, success or failure, in this case either a sunk shot or a missed shot. Our data are summarized in the following table: TREATMENT FREETHROWS SUCCESS 54 FAILURE 46 TOTAL 100
p = 0.54 The second assumption is that p is constant, if p increases the shooter is improving with practice, if p decreases the shooter is either tired or bored. Our values of p are within 0.04, which we consider close enough to be Bernoulli trials; there is little evidence of improvement or decline. SUCCESS 54 FAILURE 46 TOTAL 100

FIRST HALF SECOND HALF

p1 = 0.56 and p2 = 0.52. The third assumption is that the trials are independent of each other. The previous outcome doesnt inuence the current outcome, i.e. there is no memory. PREVIOUS TRIAL SUCCESS FAILURE CURRENT TRIAL SUCCESS FAILURE 53 46

TOTAL 99

p1 = 0.54 and p2 = 0.53. Our p values are 0.54 and 0.53. These are very close and show there is very little dependence on the previous action. The free throw shooter shot almost exactly the same after a success as a failure. 7
Interval Calculations 90% Condence Interval x = 54, n = 100, p = 0.54 and q = 0.46. 0.54 1.645 0.54(0.46) = 0.54 0.082. 100
0.54 + 0.082 = 0.622 and 0.54 0.082 = 0.458. The condence interval is [0.458, 0.622]. 80% Prediction Interval m = 100, n = 100, x = 54, p = 0.54, and q = 0.46. The point prediction is m = 100(0.54) = 54. p The prediction interval is 54 1.282 54(0.46) 1 + (100/100) = 54 9.04 = [45, 63]. Later in the day, we performed 100 more trials and Ena obtained 54 successes. Both the point and interval predictions were correct.

Model Project Number 5: Darts
Katie Ledvina and Carrie Skrip As a dart-throwing junkie, I convinced Carrie that studying my accuracy in throwing darts would be an interesting subject to study for our statistics project. Carrie agreed, since there had been many nights where we have plugged quarters into the dart machine trying so desperately to inate our egos over a game of cricket. Also, since darts is a very common activity among bar dwellers, we thought that it may be an interesting topic overall. In order to set up a xed environment, we used the electronic dartboard that I have in my apartment. I was designated dart thrower for both rounds, and Carrie kept track of all the data we collected. I stood a set distance from the board, marked by black tape, and used the same darts (accidently taken from a bar in Fond du Lac) throughout the entire experiment. The darts were thrown in a series of three, until I had made 100 throws. The red and green areas on the board were designated successes, (areas that count for doubles, triples, and bulls eye), and the black and white areas were designated failures, (areas that only count for single points). The results of the rst 100 throw were as follows: FFFFF SFFSF FSFFF FFFSS SFFFF FFFSF FSSFS FSSFF FFSFF FFSFF FFFSF FFFFF FFSFS FFSFF FFSFF SSSFS FFFFS FFFFS SSSSF FFFFF First Half Second Half Total Success 29 Failure 71 Total 100
p1 = 14/50 = 0.28 and p2 = 15/50 = 0.30. Therefore, there is evidence that Katies ability improved during the course of the study, but not by very much. Next, Previous Success Failure Total Current Success 29 Failure 70 Total 99
p1 = 10/29 = 0.34 and p2 = 19/70 = 0.27. Therefore, p1 > p2. This means that there is evidence that Katies performance was better after a success rather than after a failure. Do we have Bernoulli trials? 1. Each trial must result in 1 of 2 possibilities? YES 2. The probability of success p remains constant? The difference between p1 and p2 is only 0.02; thus, this assumption seems reasonable. 3. Are the trials independent? The difference between p1 and p2 is only 0.07; thus, this assumption seems reasonable.
On the assumption that we do have Bernoulli trials the following analysis was applied. 90% Condence interval: p z pq /n = 0.29 1.645 0.29(0.71)/100 = [0.22, 0.36]. Point prediction: m(x/n) = 100(29/100) = 29. 80% prediction interval: m z mq 1 + m/n = 29 1.282 29(0.71) 1 + 100/100 = [20, 37]. p p After this analysis we then performed another 100 trials and compared that data to the prediction interval from our prior trials. Katie threw 21 successes that did not match the point prediction. However, the 21 successes did fall into the 80% prediction interval.

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