Choosing an appropriate statistical test
[25/100%]Explain why the proposed test is wrong and identify the correct test:
My question is "Does playing on home ice in the National Hockey League (NHL) give the home team a better chance to win?" I will choose six NHL teams and, for each team, use a Z single-sample categorical test of the null hypothesis that the chance of winning at home is equal to one half.
Before coming to college, a student watched videotapes of a seminar, "Where there's a will, there's an A," which purports to show scientifically supported strategies for guaranteeing A grades. One strategy is to sit in the front of the class. To test this theory, a statistics student asked 50 randomly selected college students to write down their grade point average (GPA) and to indicate where they typically sit in large classrooms.
| Number of Students | Average GPA (12-Point Scale) | |
|---|---|---|
| In the very front | 5 | 10.94 |
| Toward the front | 21 | 9.38 |
| In the middle | 11 | 9.38 |
| Toward the back | 10 | 8.37 |
| In the very back | 3 | 10.20 |
The student used a difference-in-means test to compare the average GPA of the 26 students who sit either in the very front or toward the front with the average GPA of all 50 students. What crucial assumption is violated by this procedure? What should he have done instead?
After examining the data showing that college students who sit near the front of the classroom have relatively high GPAs, he concluded, "I would certainly advise a freshman to consistently sit in the front row in every single class." Provide a very different interpretation of his data.
Explain why the proposed test is wrong and identify the correct test:
I used a chi-square test to look at the relationship between high school graduation rates in California counties and the per capita income in the past 12 months in these counties. Here are my results for the first four counties alphabetically in California:
| Alameda | Alpine | Amador | Butte | |
|---|---|---|---|---|
| Graduation rate | 86 | 91 | 88 | 87 |
| Per capita income ($) | 35,434 | 27,135 | 26,969 | 23,556 |
Graduation rate is the percent of persons age who graduated from high school and per capita income is annual income (in 2012 dollars) for the years 2008 through 2012. The chi-square value is 8.58 and the -value is .0353, which means we would reject the null at the level, but not at the level.
A study of 15 stocks that were wildly popular in the 1970s collected these data: (1) the December 1972 price earnings ratio (P/E) and (2) the annual percentage rate of return (R) from 1973 through 2000. What statistical test would you use to see if stocks with high P/E ratios tended to have low annual returns? (You do not have to actually do the test, just identify it clearly.) Be sure to identify the null hypothesis and how you would test it.
| 1972 P/E | R | ||
|---|---|---|---|
| Polaroid | 90.7 | -6.07 | |
| McDonald's | 85.7 | 11.85 | |
| Walt Disney | 81.6 | 10.58 | |
| Avon Products | 65.4 | 6.30 | |
| Kresge (now Kmart) | 54.3 | 1.20 | |
| Simplicity Pattern | 53.1 | -2.01 | |
| Xerox | 48.8 | -1.99 | |
| Eastman Kodak | 48.2 | 2.68 | |
| Coca-Cola | 47.6 | 14.64 | |
| IBM | 37.4 | 8.65 | |
| Procter \ | Gamble | 32.0 | 12.27 |
| PepsiCo | 29.3 | 16.17 | |
| General Electric | 26.1 | 16.85 | |
| Philip Morris | 25.9 | 18.00 | |
| Gillette | 25.9 | 14.89 |
Here are several situations where there is an incorrect application of statistical reasoning. Explain briefly what is wrong and why it is wrong.
A random sample of 100 students was asked if dining-hall food is better or worse this year than last year. A difference-in-proportions test compared the who chose "improved" with the who chose "worsened."
A one-sample t-test rejected the null hypothesis that the sample mean is 25.
A matched-pair t-test reported that the results are statistically significant at the level because the p-value is .95.
A researcher calculated the price-earnings (P/E) ratios of the 30 stocks in the Dow Jones Industrial Average, as of January 1, 2017. A difference-in-means test was then used to compare the average price increase in 2017 for the 15 stocks with the lowest P/Es with the average price increase of all 30 stocks.
For which of the following tests do we need equal sample sizes?
Difference-in-means
Difference-in-proportions
Binomial
Chi-square
Regression
Identify the most appropriate statistical test. You do not need to show any formulas, just identify the test, for example, "difference-in-means t-test":
The average college student got less than 8 hours of sleep during the past 24 hours.
The average price of Hollister clothing is less than the average price of essentially identical AF clothing.
People are more likely to have high blood pressure if they eat lots of dairy products and drink very little citrus juice.
Taller male college students get better grades than do shorter male students.
Horseshoe pitchers are more likely to throw a double-ringer if they threw a double-ringer in the previous inning than if they didn't throw a double-ringer.
Identify the most appropriate statistical test. You do not need to show any formulas, just identify the test, for example, "difference-in-means t-test":
Daily trading volume tends to increase in December for stocks that have suffered large price declines between January and November.
GPAs tend to be higher for the players on women's Division III college basketball teams than for the players on men's Division III college basketball teams.
Holding earnings, risk, and growth rates constant, companies that pay higher dividends tend to have higher stock prices.
The prices of the 30 stocks in the Dow Jones Industrial Average tend to increase between the close of trading each day and the opening of trading the next day.
Japan's soccer goalie is more likely to dive to the left than to the right on penalty kicks.
Identify the most appropriate statistical test. You do not need to show any formulas, just identify the test, for example, "difference-in-means t-test":
Professional poker players tend to play looser after they lose a big pot than after they win a big pot.
The incumbent party's share of the presidential vote depends on the unemployment rate and the rate of inflation.
A company's stock price tends to go up on the Monday after it has an ad shown during the Super Bowl.
Baseball batting averages tend to increase over the course of a season.
Adult women are more likely to live in different ZIP codes from where they were born if their mother is a first-generation immigrant.
Identify the most appropriate statistical test. You do not need to show any formulas, just identify the test, for example, "difference-in-means t-test":
Time series data were used to predict interest rates based on the rate of inflation and the government deficit.
The daily returns on a professionally managed stock portfolio were compared to the daily returns on the SP 500 index during the year 2000-2020.
"As January goes, so goes the year" is an old stock market adage. Data were collected for 1970-2020 on the SP 500 return in January of each year and the SP 500 return for February through December of that year.
In golf, each hole is assigned a par score (typically between 3 and 5) that a good golfer should be able to achieve. A study of several professional golf tournaments recorded how often a player's score on a hole was below-par, par, or above-par, and how often this player's score on the next hole was below-par, par, or above-par.
To test a mentalist's claim to be able to influence dice rolls, two standard six-sided dice were rolled 200 times and the number of doubles was recorded.
Identify the most appropriate statistical test. You do not need to show any formulas, just identify the test, for example, "difference-in-means t-test":
A study of 463 stock splits between January 1, 2007 and December 31, 2016, compared the return on these stocks on the day the split was announced to the return on the SP 500 index that day.
A study predicted the number of hours a person spent on volunteer work each month based on the person's sex, race, age, and income.
A study compared average starting salaries for recent college graduates hired by Facebook, Amazon, Netflix, Google, and Apple.
The number of children a woman has (none, or more) is related to the number of children her mother had.
A study used data for all 58 California counties to compare the difference between the 2009 and 2017 divorce rate with the difference between the 2009 and 2017 unemployment rate.
Identify the most appropriate statistical test. You do not need to show any formulas, just identify the test, for example, "difference-in-means t-test":
Data for 48,440 adult patients infected with COVID-19 were used to compare the average age of those who died with the average age of those who did not die.
The infected patients were divided into three physical activity groups (consistently inactive, some activity, and consistently active) and the researchers compared the frequency with which people in each physical activity group were hospitalized.
The number of infected male and female patients were compared to see if infected patients are equally likely to be male or female.
The infected patients were divided into four age groups (, 60-69, 70-79, and ) to see if there was a relationship between age and physical activity.
Each patient's probability of dying was predicted based on the patient's physical activity category and 25 other possible confounding variables.
Identify the most appropriate statistical test. You do not need to show any formulas, just identify the test, for example, "difference-in-means t-test":
A study of injuries suffered by varsity athletes during the years 1980-1995 obtained these data:
| Females | Males | |
|---|---|---|
| Basketball | 214 | 344 |
| Soccer | 155 | 236 |
| Swimming | 161 | 34 |
| Tennis | 82 | 82 |
| Track | 151 | 233 |
Data for 116 California communities were used to see how the percent change in home prices between 2005 and 2010 was affected by each county's percent change in home prices between 2000 and 2005 and each county's ratio of the median home price in 2005 to median rent in 2005.
A researcher looked at 50 stocks that had been removed from the Dow Jones Industrial Average and, in each case, identified whether, after the substitution, the stock that was removed did better or worse than the stock it replaced it.
Daily stock return data were used to see whether a portfolio of companies removed from the Dow Jones Industrial Average did better than a portfolio of the companies that replaced them.
Survey data were collected from 120 college students on the average time they went to bed each night during the previous semester and their GPA that semester.
Identify the most appropriate statistical test. You do not need to show any formulas, just identify the test, for example, "difference-in-means t-test":
More goals are scored in the second half of soccer games than in the first half.
A researcher investigated discrimination against women by a certain employer by comparing male and female salaries, taking into account education, years of experience, and other factors the employer claimed affected salaries.
Stocks with high price-earnings ratios at the peak of the dot-com bubble did worse than other stocks over the next 20 years.
A study compared male and female ratings (on a scale of 1-10) of Coke and Diet Coke.
A researcher investigated whether Jews are able to postpone their deaths until after Passover by counting the number of Jews who died during each of the 8 weeks surrounding Passover: 4 weeks before, 3 weeks before, 2 weeks before, 1 week before, 1 week after, 2 weeks after, 3 weeks after, and 4 weeks after.
Identify the most appropriate statistical test. You do not need to show any formulas, just identify the test, for example, "difference-in-means t-test":
A study compared the daily returns from a portfolio of the stocks recommended by the book, Good to Great, with the SP 500 index.
A study investigated whether the fluctuations in the daily returns for ten stocks chosen by Fortune as the "most admired" could be explained by the overall stock market, each company's dividends, and each company's earnings.
A researcher investigated a strategy of betting against professional football teams that won the previous week by counting the number of bets won and lost using this strategy during the years 2000-2020.
A study compared the average income today of people who graduated 20 years ago from two different colleges.
A researcher investigated the hot hands phenomenon by tabulating how often professional bowlers followed a strike, spare, or open frame with a strike, spare, or open frame.
Identify the most appropriate statistical test. You do not need to show any formulas, just identify the test, for example, "difference-in-means t-test":
Oak trees are more likely to survive hurricanes than are pines trees or spruce trees.
The price of undeveloped land on the outskirts of a city depends on how close it is to the city center, and whether there are nearby water and electrical connections.
Identical brand-name products cost less at Costco than at Target.
The average age at death is the same for people with common and uncommon names.
A study compared daily stock-market returns on the New York Stock Exchange to the amount of cloudiness that day in New York (measured on a scale from 0 to 100).
Identify the most appropriate statistical test. You do not need to show any formulas, just identify the test, for example, "difference-in-means t-test":
Data collected by the Center for Coastal Studies found that female whales with an interval of less than 3 years between calving had 22 sons and 20 daughters. For whales calving at intervals of 3 or more years, there were 16 sons and only 4 daughters.
The ad revenue generated on a travel company's web page does not depend on whether the main page shows a picture of a beach, forest, or city.
A study of severe heart attacks suffered by German workers (either on or off the job) grouped the data according to the day of the week when the heart attack occurred.
The SAT scores and high school GPAs of 500 students were used to predict their first-year college GPAs.
In 1977, the US Supreme Court noted that persons with Spanish surnames constituted of the population of Hidalgo County, Texas, but only 339 of 870 recent jurors in this county.
Identify the most appropriate statistical test. You do not need to show any formulas, just identify the test, for example, "difference-in-means t-test":
Math majors who play varsity sports have higher GPAs than do other math majors.
A baby's birth weight is related to the mother's pre-pregnancy weight.
A study compared the income of immigrant mothers and their daughters when each daughter was the same age as her mother at the time of the daughter's birth (e.g., if the daughter was born when her mother was 28, the study compared the mother and the daughter's income when each was 28).
The heights of Major League Baseball players do not depend on the position they play: catcher, pitcher, infield, or outfield.
The daily percentage change in the price of gold is independent of the percentage change the day before.
Identify the most appropriate statistical test. You do not need to show any formulas, just identify the test, for example, "difference-in-means t-test":
The sex of a mother's third child is independent of the sexes of her first two children.
An increase in the rate of inflation increases the interest rate on Treasury bills by the same amount.
People who smoke cigarettes are more likely to die of lung cancer than are nonsmokers.
A study tested the theory that two main determinants of stock prices are the unemployment rate and interest rates.
The daily returns from a portfolio of ten AI-powered mutual funds were compared to the SP 500 from January 1, 2020, through December 31, 2024.
Identify the most appropriate statistical test. You do not need to show any formulas, just identify the test, for example, "difference-in-means t-test":
A black-box algorithm predicts whether the Dow Jones Industrial Average will go up or down each day.
A study compared the off-season and on-season GPAs of college students who play a varsity sports in either the spring or fall (but not both).
A study compared the success rates of soccer penalty shots taken to the left, right, and middle of the goal.
At-risk patients are less likely to have a fatal heart attack if they take aspirin daily instead of a placebo.
A study of the effect of quarries on home prices compared the prices of homes located at different distances from a quarry, taking into account each home's square footage, lot size, number of bedrooms, number of bathrooms, and the age of the home.
Identify the most appropriate statistical test. You do not need to show any formulas, just identify the test, for example, "difference-in-means t-test":
Home prices in Omaha depend on square footage, number of bedrooms, and number of bathrooms.
Thirty people label the pictures of Joe Biden shown on the front pages of the New York Times and the New York Post on the day after the 2020 presidential election as "favorable" or "unfavorable."
Essays written by 100 students in a college history class are graded on a scale of 1-100 by the professor and by an AI algorithm.
A researcher wanted to see if natural births are more likely at certain times of the day. She divided the day into 24 one-hour periods and recorded the number of births in each one-hour period.
A company's stock price usually rises after it announces the departure of its CEO.
Identify the most appropriate statistical test. You do not need to show any formulas, just identify the test, for example, "difference-in-means t-test":
Data were collected on the market values of 25 undamaged South Napa homes 6 months before and after the 2014 South Napa earthquake to see what effect, if any, the quake had on home values.
Daily stock returns in 2020 were collected for the SP 500 index of stocks and for a portfolio of the publicly traded stocks identified by Glassdoor on December 10, 2019, as among the 50 best companies to work for.
It was estimated that of the people living in Dublin have Brady as a last name and that 8 of the 999 people treated in Dublin hospitals for bradycardia (a slower than normal heart rate) have Brady as a last name.
Thirty-six subjects read a paragraph about an elderly man named George. For 18 people, the paragraph described him as competent; for the other 18, he was described as incompetent. After reading the story, each subject indicated how warm and friendly George is on a scale of 1 (not at all) to 9 (very).
After its development, the Pfizer mRNA vaccine was tested through a randomized controlled trial of 37,586 participants, with 18,801 people in the treatment group given the vaccine and 18,785 people in the control group given an injection of a saline placebo. There were 8 cases of COVID-19 in the treatment group, compared to 162 in the control group.
Identify the most appropriate statistical test. You do not need to show any formulas, just identify the test, for example, "difference-in-means t-test":
There are more motorcycle accidents on Friday nights with a full moon than on Friday nights with no moon.
A researcher looked at daily returns in the stock market over the past 20 years to see if returns tend to be relatively good or bad on certain days of the week.
A researcher looked at the performance of stocks with clever ticker symbols (such as MOO for United Stockyards) by comparing the daily returns for a portfolio of clever ticker stocks with the SP 500.
A survey of 100 college students compared the percentage who were first-born or only children to the national percentage of .
A survey of 150 college students compared the number of firstgeneration, second-generation, and other college attendees with their choice of a STEM major or a non-STEM major.
Identify the most appropriate statistical test. You do not need to show any formulas, just identify the test, for example, "difference-in-means t-test":
A researcher investigated the socioeconomic mobility of California women by using data on each woman's income, her parent's income, whether or not her mother was born in the United States, and whether or not her father was born in the United States.
Twenty-four students were asked to read a short article and when they were done, were asked to guess how long it took them to read the article; 21 of 24 students overestimated the time.
The time it took each of the 24 students in part (b) to read the article was compared to his or her guess.
An Internet consulting company tested commercial web page layouts by randomly directing users to different pages; for example, a background that is a light blue, light red, light green, or white. The company then measured the click-through rate for each layout (the fraction of users who click on a link that gives them more information about a product).
A researcher compared the price/earnings (P/E) ratios of 50 popular ("Nifty 50") stocks on December 31, 1972, and the percentage returns on each of these 50 stocks from 1973 through 2022.
If a company is considering changing the main color on a web page from, say, blue to red, green, or purple, in order to increase the number of users who "click through" to another page, BestWeb sets up four pages using these colors. When a user goes to the page, a random number generator is used to take the user to one of the four pages and BestWeb records the click-through rate (fraction of users who click through). If there is a statistically significant difference, the client is advised to use the color with the highest click-through rate.
What is the appropriate statistical test?
When a client follows BestWeb's advice to change the predominant color of its web page, the client often finds that the actual increase in the click-through rate is not as large as observed in BestWeb's experiment. How would a statistician explain this?