Miscellaneous

The value of participating in college debate was demonstrated by comparing the GPAs of debaters with non-debaters.

Correlations between a person's BMI and consumption of 100 different foods found that BMI was reduced by the consumption of apple juice.

A 2021 study of the ten largest mutual funds found that they had outperformed the SP 500 by more than two percentage points a year over the period 2000-2020.

A 2020 cross-sectional study of the relationship between the average female unemployment rate and the daytime low temperature in 29 countries during the years 1997 through 2014 found a statistically significant negative correlation.

An unusually large number of tweets of the word "happy" by Donald Trump tend to be followed by an increase in the level of the Dow Jones Industrial Average 3 days later.

Heart disease fatalities were reported for the 8 heart-disease categories (out of 17 total categories) in which Asian-Americans have an above-average death rate on the fourth day of the month.

A regression model found a statistically significant relationship between household spending and wealth after income was dropped as an explanatory variable.

The benefits of studying French were demonstrated by the fact that high school students who had taken French classes scored 80 points higher, on average, than other students on the Reading/ Writing SAT.

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 $57 \%$ who chose "improved" with the $43 \%$ 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 $5 \%$ 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.

Difference-in-means

Difference-in-proportions

Binomial

Chi-square

Regression

If R-squared in a simple regression model is positive, then the correlation $R$ is positive.

In a chi-square test, if $\chi^{2}=1$, then $\mathrm{p}=0$.

$\mathbb {P}\left(\mathrm{A} \text{ or } \mathrm{B}\right)$ is never equal to $\mathbb {P}\left(\mathrm{A}\right)+\mathbb {P}\left(\mathrm{B}\right)$.

The variance of the sample mean of $X$ cannot be larger than the variance of X.

The variance of a binomially distributed variable $X$ cannot be larger than its mean.

If two sets of data have the same mean and standard deviation, their histograms are identical.

The median is never outside the interquartile range.

If A and B are independent, then $\mathbb {P}\left(\mathrm{A} \text{ or } \mathrm{B}\right)=\mathbb {P}\left(\mathrm{A}\right)+\mathbb {P}\left(\mathrm{B}\right)$.

A normal distribution is symmetrical about its median.

The expected value of a probability distribution is the most likely outcome.

Since the p -value is .089, I accept the null hypothesis.

If $X$ 's mean is larger than $Y$ 's mean, then $X$ 's median is larger than Y's median.

In a box plot, an outlier is never inside the box.

Multicollinearity is when a dependent variable is correlated with the explanatory variables.

The explanatory variables in a multiple regression model have little effect on the dependent variable if R -squared is low.

If R-squared is not high enough, you should add more explanatory variables.

If there is a multicollinearity problem, you should drop some explanatory variables.

If the null hypothesis is true, the p-value is more likely to be less than .10 than less than .05.

The least squares line goes through the sample means of X and Y.

The probability of A or B is equal to the probability of A plus the probability of $B$.

A difference-in-means test assumes equal sample sizes.

A difference-in-means test assumes equal standard deviations.

A normal distribution is better than a binomial distribution for a single-sample test of a probability.

A multiple regression model assumes that the explanatory variables are uncorrelated.

A multiple regression model assumes that the explanatory variables are normally distributed.

If I flip a fair coin until I obtain a head, on average, it will take me two flips.

I used a normal approximation to the binomial distribution because the central limit theorem tells us that for large sample sizes, the distribution of the success probability p is approximately normal.

If the expected value of the change in income is zero, then the change is equally likely to be positive or negative.

The probability of 50 heads when a coin is flipped 100 times is larger than the probability of 5 heads when a coin is flipped 10 times.

To investigate the relationship between interest rates and stock prices, I tested the null hypothesis that there is an effect against the alternative hypothesis that there is no effect.

An ANOVA F-test and a difference-in-means t-test give the same p -value if there are two samples.

Because the p-value is .01, I rejected the null hypothesis that the sample mean is -2.

Because the p -value is larger than .05, we can reject the alternative hypothesis at the $5 \%$ level.

The F-value will be negative if all the sample means are negative.

I rejected the null hypothesis at the $5 \%$ level because the p-value is larger than .95.

If I roll a six-sided die until I obtain a 6, it will, on average, take me six rolls.

The average absolute deviation is less sensitive to outliers than is the standard deviation.

The total area of the bars in a histogram is equal to 1.

The central limit theorem assumes that $X$ has a binomial distribution.

I used a t-test instead of a Z-test because the sample size was so large.

The central limit theorem explains why a binomial distribution with $\mathrm{p}=.9$ converges to a normal distribution as n increases.

For a normal distribution, there is a .95 probability of being within two standard deviations of the median.

The width of the box in a box plot is equal to the interquartile range.

The probability of four 2 s when six dice are rolled can be determined by the binomial distribution.

A t -value can never be negative.

A significance test that is significant at the $1 \%$ level is also significant at the $5 \%$ level.

The ANOVA F-statistic can be used to test the null hypothesis that the sample means are equal.

If the chi-square value is 0, then the p -value is 0.

If the null hypothesis is $\mathrm{p}=.5$, you can use a one-sided p -value once you know whether $\mathrm{x} / \mathrm{n}$ is larger or smaller than .5.

In a simple regression, R -squared is equal to the correlation coefficient squared.

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.

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.

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.

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.

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, $1,2,3,4$ 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.

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 ($<60$, 60-69, 70-79, and $>79$) 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.

A study of injuries suffered by varsity athletes during the years 1980-1995 obtained these data:

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.

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.

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.

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).

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 $79 \%$ of the population of Hidalgo County, Texas, but only 339 of 870 recent jurors in this county.

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.

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.

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.

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.

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 $.36 \%$ 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.

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 $40 \%$.

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.

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.

Mean

Median

Standard deviation

Interquartile range

The average actual score (59.1) was lower than the average predicted score (74.9).

There was a. 65 correlation between the predicted and actual scores.

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?

"The standard deviation is larger for prime properties because of the greater diversity of more expensive homes in prime London neighborhoods as compared with the Liverpool housing market."

"The luxury ratio of prime London with Liverpool represents the average excess amount, in percentage terms, that is paid in prime London relative to Liverpool for a property. A luxury ratio of 150, for example, signifies that the price level of prime property is $50 \%$ greater than the price in Liverpool."

"We can also observe that 2005 was the right moment to buy prime property in London if one meant to do so. Premiums paid in prime London with respect to Liverpool greatly increased after 2005."

What fraction of the null hypotheses tested by this researcher will be rejected?

What fraction of all the null hypotheses that this researcher rejects are in fact true?