Chi-Square Tests

It has been argued that when people are asked to choose a number, say from 1 to 4 or 1 to 10, they tend to choose an "average number" that is in the middle rather than an "extreme number" on either end. To test this theory, a professor did a mock ESP experiment, in which 40 students were asked to guess which one of four cards, numbered 1-4, the professor had chosen randomly. Are the patterns in these data substantial and statistically persuasive?

In the game Roshambo (rock-scissors-paper), two players simultaneously show a fist (rock), two fingers (scissors), or an open hand (paper). Rock beats scissors, scissors beat paper, and paper beats rock. A researcher played this game against 120 people, recording the initial move of each opponent. Use these data to test the null hypothesis that rock, scissors, and paper are used equally often on the initial move: 27 rock, 54 scissors, and 39 paper. Are the observed differences substantial?

When the Titanic sank in 1912, it was carrying 1,315 passengers: 402 adult females, 805 adult males, and 108 children. Of the 498 passengers who were saved, there were 296 adult females, 146 adult males, and 56 children. In which group was the fraction saved disproportionately large? Disproportionately small? Do these data reject at the $1 \%$ level the null hypothesis that a person's chances of being saved did not depend on the group he or she was in?

A statistics professor flipped a fair coin 10 times and asked each of the 24 students to guess whether the flip was a head or tail. Each student then found the longest streak in their guesses; for example, in these data H T H H T H T T T H, the longest streak is 3. The results were:

What pattern do you see in these results? Are the deviations from the expected values substantial and statistically persuasive? If so, how would you explain these patterns?

A study of major league baseball (MLB) managers from 1871 through 2007 found that 540 managers played professional baseball before becoming a manager, with the primary positions grouped as follows: pitcher (51), catcher (116), infielder (262), and outfielder (111). Use these data to test the null hypothesis that the respective probabilities are 1/9 pitcher, 1/9 catcher, 4/9 infielder, and 3/9 outfielder. What patterns do you see in your data? Are these patterns substantial and statistically persuasive?

The five positions on a basketball team are shown below. As of 2012, 167 of the 299 head coaches in the NBA played in the NBA. Their playing positions:

Is there substantial and statistically persuasive evidence against the hypothesis that coaches are equally likely to come from each of these five positions?

A study looked at the positions played by the 70 managers in the top five professional soccer leagues during the 2022 season who had played professional soccer:

Assuming that $3 / 11$ of all players are forwards, $3 / 11$ midfielders, 4/11 defenders, and 1/11 goalies, what patterns do you see in your data? Are these patterns substantial and statistically persuasive?

When 100 MIT MBA students were offered these three subscription choices to The Economist, 16 chose the online subscription, none chose the print subscription, and 84 chose the combination subscription:

When 100 other students were offered just two choices, 68 chose the online subscription and 32 chose the combination subscription:

Is this observed difference in the online and combination subscription choices between these two offers substantial and statistically persuasive?

A student asked a random sample of 54 of her classmates whether they were nearsighted and whether their grade point average (GPA) was above or below B+:

One hundred students were asked this question:

You have a coach ticket for a flight from New York to London. Because the flight was overbooked, randomly selected passengers will be allowed to sit in open first-class seats. You are the first person selected. Would you rather sit next to: (a) Joe Biden; (b) Jill Biden; or (c) Michael Jordan?

Here are the results:

An online college statistics class asked students to apply a chi-square test to these observed and expected values:

Why is this question flawed?

Seventy-two randomly selected college students were asked to taste three unlabeled chocolate chip cookies and identify the one they liked best: 13 students chose Chips Ahoy; 12 chose Lady Lee, and 47 chose supermarket bakery cookies. Are these differences substantial and statistically persuasive? What is your null hypothesis?

On a full rack of billiard balls, 99 break shots resulted in 54 sunk balls, allocated among the 15 balls as follows:

Dividing the rack into three categories (corner, adjacent, and other), is there a substantial and statistically persuasive relationship between a ball's position in the rack and its likelihood of being sunk on the break?

A Russian statistician named Ladislaus Bortkiewicz investigated whether the number of soldiers killed by horse kicks was randomly distributed among Prussian cavalry corps. He collected data for 122 deaths over a 20 -year period in 12 cavalry corps. Using the Poisson distribution, he calculated the expected number of cases in which a corps would have no deaths in a single year (108.67), the expected number of cases in which a corps would have a single death (66.29), and so on. He compared these expected numbers to the actual number of cases. It has been suggested that the correspondence between the predicted and observed instances is too good to be true. How would you test this?

A study of first drafts of student papers tabulated how frequently letters were used and the frequency with which typographical errors occurred. For example, "Firday" is an i typo and an r typo; "Froday" or "Frday" is an i typo; and "Friuday" is a u typo. The data were grouped according to the row on the keyboard:

Calculate the expected number of errors for each of these four categories, assuming that the 75 errors are distributed in accord with the frequency with which letters are used. Are the differences between the expected and actual number of errors substantial and statistically persuasive?

A researcher suspected that a disproportionate number of people might be born in some months (like November, which is 9 months after Valentine's Day). Use the following data on the birth months of

148 college students to test at the $5 \%$ level the null hypothesis that all birth months are equally likely:

A university study examined the relationship between a woman's shoe size and whether the woman frequently experienced foot pain. Are the observed differences substantial and statistically persuasive?

The Mandarin, Cantonese, and Japanese pronunciation of "four" and "death" are almost identical, and many Chinese and Japanese people consider the number 4 to be unlucky. Californians have some say in the last four digits of their telephone numbers, and a study of the last four digits of the telephone numbers of 1984 Chinese and Japanese California restaurants counted the number of times the digits 0-9 appeared. Are the differences among the digits substantial and statistically persuasive?

It has been argued that people may be able to postpone their deaths until after the celebration of an important event. Here are data for Jewish members of Sinai Memorial Chapel in San Francisco who died within 4 weeks of Passover between January 1, 1987, and December 31, 1995. Do these data provide substantial and statistically persuasive support for the theory?

A student observer attended eight different college classes and recorded the number of female and male students in each classroom, and the sex of each student who made a spoken remark, question, or comment directed at the professor. She found that there were 88 females and 107 males in these classes and a total of 276 student exchanges with the professor, of which 138 were made by females and 138 by males. Are these observed differences substantial and statistically persuasive?

A survey conducted by the National Opinion Research Center asked 756 randomly selected Americans if too much or too little money was being spent on "welfare," while a random sample of 761 Americans were asked if too much or too little money was being spent on "assistance to the poor." Are the differences in the results substantial and statistically persuasive?

In order to see whether women are more successful at single-sex or coeducational colleges, samples of women attending a women's college and a coeducational college were asked, "Do you feel you are successful at your college?" The results were as follows:

The researcher explained that "A chi-square test was used to see if the results are statistically significant. We assume that a positive answer is a valid measure of true success, so that the null hypothesis is that the probability equals 0.5, because there is a 50 -percent chance of agreeing that either yes, one is successful or no, one is not successful." The observed values were compared to these expected values:

Explain why this procedure is not persuasive, and then make an appropriate statistical test.

Eighty-five randomly selected college students were asked if they had a serious romantic relationship in the past 2 years and, if so, to identify the month in which the most recent relationship began.

A chi-square test was used to test the null hypothesis that each month is equally likely for the beginning of a romantic relationship. Identify three distinct errors in the conclusion (do not check the math):

The expected value for each month is $85 / 12=7.083$. The chi-square value is 15.66 and the two-sided p-value is .154. Therefore, the deviations between the observed and expected values are too small to be explained plausibly by random sampling error. The results are statistically substantial because there are more than 5 degrees of freedom.

A professor showed 56 students four playing cards, numbered 1, 2, 3, and 4 ; shuffled them behind his back; and then selected one of the cards. While he looked at the selected card, the students tried to read his mind and wrote down their best guess as to which card was selected. Here are the results:

What is wrong with this calculation of the chi-square statistic and why does it matter?

A researcher looked at male and female choices between Coke and Diet Coke and obtained a chi-square value of .0383: "This means that I can reject the null hypothesis that men and women are equally likely to prefer Coke or Diet Coke, since 0.0383 is less than 0.05." What is incorrect about this conclusion?

Two professors looked at the birth months of 76 major league baseball players who committed suicide. They calculated the adjusted number of suicides in each birth month by dividing the number of suicides in that month by the total number of players with that birth month and multiplying by 1,000. For example, there were a total of 638 players with January birth months; so, the adjusted January number is $1,000(6 / 638) =9.4$, rounded off to 9. The expected value in each month was 126/12= 10.5 and their calculated chi-square value was

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:

Graduation rate is the percent of persons age $25+$ 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 $p$-value is .0353, which means we would reject the null at the $5 \%$ level, but not at the $1 \%$ level.

Two researchers collected birthday and deathday data for 120 randomly selected deceased celebrities. These data were divided into three categories: deaths that occurred during the 30 days preceding the birthday; deaths that occurred on the birthday or during the 29 days following the birthday; and deaths on other days. A chi-square statistic was used to test the null hypothesis that a person's deathday is not related to his or her birthday:

What fundamental problem do you see with this analysis?

A researcher surveyed 104 of the 1,579 students at a college to see if religious beliefs varied by grade level, suspecting that students who attend college become less religious between their freshman and senior years:

He did not find a statistically significant relationship, so he "used the sample mean to estimate the population mean;" for example, (1,579/104) $11=167$. His chi-square value was now 102.29, which is statistically significant at the $5 \%$ level. What is the flaw in his procedure?

For several years, the manufacturer of plain MM's claimed that the overall percentages of the candies of different colors were as follows: $30 \%$ brown, $20 \%$ red, $20 \%$ yellow, $10 \%$ blue, $10 \%$ green, and $10 \%$ orange. A randomly selected 17.6 ounce bag of plain MM's yielded the following data: $34 \%$ brown, $18 \%$ red, $21 \%$ yellow, $12 \%$ blue, $7 \%$ green, and $8 \%$ orange. The researchers used this chi-square statistic:

The p -value was .78. Their conclusion: "The probability of every color matching the null hypothesis was fairly high, and could not be rejected at the $5 \%$ level."