One-Sample Tests
[21/0%]Find someone who claims to have extrasensory perception (ESP) and test this claim.
Morningstar and other Internet websites have data on the total returns (dividends plus capital gains) for mutual funds in various fund categories (e.g., US Equity Small Value). Pick a category within US Equity Funds and select a horizon (e.g., 5 years). Now determine the p -value for a test of the null hypothesis that these data were drawn from a population with a mean equal to the return on the SP 500 index over this horizon. (Make sure that your SP 500 return is not just the change in the SP, but the SP total return, dividends plus capital gains.) Also use these data to determine the p-value for a test of the null hypothesis that a randomly selected mutual fund has a .5 probability of doing better than the SP 500 index.
A high school basketball coach said that a missed free throw by a right-handed shooter is more likely to bounce to the right, while the reverse is true of a left-hander. To investigate this claim, find five avid basketball players and ask each of them to shoot 100 free throws. Do not tell them the purpose of this experiment, which is to determine if a missed free throw is equally likely to bounce to the same or opposite side as their shooting hand. Use your data to calculate the twosided p -value for testing the null hypothesis that missed free throws are equally likely to bounce to either side.
Young children who play ice hockey are separated by age. In 2024, for example, children born in 2017 were placed in the 7 -year-old league and children born in 2018 were placed in the 8 -year-old league. A student with a December 11 birthday observed that children with birth dates early in the year are months older than those with later birth dates-someone born in January 2018 is eleven months older than someone born in December 2018. Because coaches give more attention and playing time to better players, this student suspected that children with early birth dates have an advantage when they are young that might cumulate over the years. To test this theory, he looked at the birth dates of 1,487 National Hockey League players and found that 934 had birth dates during the first 6 months of the year. Test his theory using data for another group of professional athletes; for example, male or female soccer players.
Calculate the percentage change in the Dow Jones Industrial Average from the close on Thursday the 12th to the close on Friday the 13th for every Friday the 13th beginning in 1980. Is the average percentage change substantial? Determine the two-sided p-value for a test of the null hypothesis that the mean percentage change is 0.
The Santa Claus Rally claims that the stock market does unusually well the week before Christmas, December 18 through December 24. A variation is that the market does unusually well the week after Christmas, December 26 through January 1. Test each of these two theories with data on the daily percentage changes in the SP 500 during at least the past 20 years.
Use a single newspaper or national newsmagazine to collect pictures of the winner of a presidential election, some printed a month before the election and an equal number printed a month after the election.
All the pictures should be from the same newspaper or magazine and be full face shots of approximately equal sizes. Do not otherwise screen the photos for being attractive or unattractive. Ask a random sample of students to pick the photo that they consider to be the most flattering and the photo that they consider to be the least flattering. Are their choices equally likely to be from the pre-election and postelection categories?
Select two prominent newspapers, magazines, or news websites with very different political reputations (e.g., CNN and Fox). Collect an equal number of pictures from each source of a presidential candidate printed a month before the election. All of the pictures should be full face shots of approximately equal sizes. Do not otherwise screen the photos for being attractive or unattractive. Ask a random sample of students to pick the photo that they consider to be the most flattering and the photo that they consider to be the least flattering. Are their most-flattering choices equally likely to be from each source? What about their least-flattering choices?
Conduct a taste test of either Coke versus Pepsi or Diet Coke versus Diet Pepsi. Survey at least 50 students who identify themselves beforehand as cola drinkers with a definite preference for one of the brands you are testing. Calculate the fraction of your sample whose choice in the taste test matches the brand identified beforehand as their favorite. (Do not tell your subjects that this is a test of their ability to identify their favorite brand; tell them it is a test of which cola tastes better.) Calculate the p-value for a test of the null hypothesis that there is a .5 probability that a cola drinker will choose his or her favorite brand.
Do more expensive cookies taste better than less expensive ones? Choose two brands of cookies that are similar in appearance but cost quite different amounts. Ask at least 50 persons to taste an unlabeled cookie from each brand and to rate each cookie on a scale of 1-10. Calculate the difference in each person's score and see if the average difference is substantial and statistically persuasive.
The pink tax says that clothing marketed as female clothing tends to cost more than identical clothing marketed as male clothing. Test this theory by going to a store (online is okay) that sells seemingly identical male and female clothing. Record the price difference for each item and see if the average difference is substantial and statistically persuasive.
The blue tax says that athletic clothing marketed as male clothing tends to cost more than athletic clothing marketed as female clothing. Test this theory by going to a store (online is okay) that sells seemingly identical male and female athletic clothing. Record the price difference for each item and see if the average difference is substantial and statistically persuasive.
The pink tax says that products marketed as female products tend to cost more than identical products marketed as male products. Test this theory by seeing whether women's shaving cream tends to be more or less expensive than men's shaving cream. Go to a store that sells seemingly identical male and female shaving cream made by the same company. Record the price difference for each shaving cream and see if the average difference is substantial and statistically persuasive. (You can use a different product if you want.)
Ask a random sample of 50 students to pick a sibling or friend who is the same sex as them. Then ask questions such as these: "Who do you think is more likely to suffer from food poisoning this year, you or [this other person]?" "Who is a better driver, you or [this other person?]" "Who is likely to live longer?" Test the null hypothesis that people are equally likely to say themselves or someone else.
Ask 100 randomly selected students to tell you whether they were the first-born (or only) child in their family. Calculate the p-value for a test of the null hypothesis that the probability that a student at your school is first-born (or only) is equal to the national percentage of .
Some people slip shoes that have shoelaces on and off without untying the shoelaces. Of those who do untie their shoelaces, some people untie the laces before taking their shoes off; others slip off their still-tied shoes and untie the laces later, when they are putting their shoes back on. What percent of the students who untie their shoelaces, either before taking them off or before putting them back on, do you think would answer "before" to this question: "When you take off shoes that have shoe laces, do you generally untie the laces before taking your shoes off or before putting them back on?" Survey 100 randomly selected students and test your prediction.
Do a matched-pair comparison of the calories in McDonald's menu items in two different countries; for example, the United States and the United Kingdom. Is the average difference substantial and statistically persuasive?
There are 20 clubs in the English Premier League (EPL). During the course of a season, each club plays each of the other 19 clubs twice, once on its home field and once away, on the opposing team's home field. There are a total of 190 unique two-team pairings. For each of these 190 games, calculate the differential between the score difference for the home game and for the away game. For example, in the 2021 season, Manchester City defeated Manchester United by a score of 4-1 in the game played at Manchester City's home field and Manchester City defeated Manchester United by a score of 2-0 in the game played at Manchester United's home field. The differential between the home difference and the away difference is 1, regardless of whether we look at it from Man City's viewpoint, , or Man United's viewpoint, . Calculate the home-minus-away differential for all 190 pairings in a recent season and test the null hypothesis that the average differential is zero.
Each team in the English Premier League (EPL) plays each of the other 19 teams twice every season-once on its home field and once away, on the opposing team's home field. Choose a recent season and determine the number of red and yellow cards that were given to each team in each game. Test the null hypothesis that the expected value of the difference between the number of cards given to a team does not depend on whether the game is played at home or away.
Persi Diaconis, a famous mathematician, claims that a spun U.S. penny will land tails about of the time. Put a clean Lincoln penny on a flat surface and hold the coin on its edge with one finger. Then flick the coin with a finger on your other hand. Repeat this experiment 100 times and test the null hypothesis that the probability that a spun penny will land tails is .80.
Choose an essay that a student has written for a school class. Ask ChatGPT or another large language model (LLM) to write an essay on the same subject. Show the two essays to a random sample of 30 students and ask each student to identify the LLM-generated essay. Use your results to test the null hypothesis that each essay is equally likely to be chosen.