13. Chi-Square Tests & Goodness of Fit

If the expected count of a category is less than 1, what can be done to the categories so that a goodness-of-fit test can still be performed?

Many municipalities are passing legislation that forbids smoking in restaurants and bars. Bar owners claim that these laws hurt their business. Are their concerns legitimate? The following data represent the smoking status and frequency of visits to bars from the General Social Survey. Do smokers tend to spend more time in bars? Use the α = 0.05 level of significance.

Does It Matter Where I Sit? Does the location of your seat in a classroom play a role in attendance or grade? To answer this question, professors randomly assigned 400 students * in a general education physics course to one of four groups. Source: Perkins, Katherine K. and Wieman, Carl E, “The Surprising Impact of Seat Location on Student Performance” The Physics Teacher, Vol. 43, Jan. 2005.

The 100 students in group 1 sat 0 to 4 meters from the front of the class, the 100 students in group 2 sat 4 to 6.5 meters from the front, the 100 students in group 3 sat 6.5 to 9 meters from the front, and the 100 students in group 4 sat 9 to 12 meters from the front.

b. For the second half of the semester, the groups were rotated so that group 1 students moved to the back of class and group 4 students moved to the front. The same switch took place between groups 2 and 3. The attendance for the second half of the semester averaged 80%. The data show the attendance records for the original groups (group 1 is now in back, group 2 is 6.5 to 9 meters from the front, and so on). How many students in each group attended, on average? Is there a significant difference in attendance patterns? Use the alpha=0.05 level of significance. Do you find anything curious about these data?

Game Boss In video games, a game boss is a powerful non-player character created by game developers as an opponent to players of the game. Suppose a game is set up where a player must defeat three bosses and the probability of defeating any boss is 0.20. Assuming each boss battle is independent, the probability distribution for the number of bosses defeated by a player is as follows:

Suppose the game is played by a random sample of 1000 players with the number of bosses defeated recorded. The results are shown below.

a. Does the distribution of defeats follow the distribution expected by the programmers? Use the alpha = 0.05 level of significance.

World Series Are the teams that play in the World Series evenly matched? To win a World Series, a team must win four games. If the teams are evenly matched, we would expect the number of games played in the World Series to follow the distribution shown in the first two columns of the following table. The third column represents the actual number of games played in each World Series from 1930 to 2019. Do the data support the distribution that would exist if the teams are evenly matched and the outcome of each game is independent? Use the α = 0.05 level of significance.

A pit boss is concerned that a pair of dice being used in a craps game is not fair. The distribution of the expected sum of two fair dice is as follows:

The pit boss rolls the dice 400 times and records the sum of the dice. The table shows the results. Do you think the dice are fair? Use the α = 0.01 level of significance.

Rationalized Lies Do people cheat or lie when the cheating or lying is not easy to identify (such as filing of taxes)? A total of 2568 college-aged subjects from various countries throughout the world rolled a single six-sided die twice. The subjects were told that the first roll counted in determining a reward and the second roll was only to determine whether the die was “working properly.” Rewards were as follows: rolling a one meant earning 1 unit of the local currency (such as \$1), rolling a two meant earning 2 units, and so on—except that rolling a six meant earning nothing. The rolling was done unsupervised (although results were secretly recorded) with the subjects free to report the outcomes of their respective rolls of the die (thereby creating an opportunity to cheat or lie about the outcome). Source: Gachter, Simon and Schulz, Jonathan, “Intrinsic Honesty and the Prevalence of Rule Violations Across Societies,” Nature (24 March 2016) 531, 496–499.

a. If individuals do not lie about the outcome of the first roll of the die, what would you expect the distribution of outcomes to be?

True or False? In Exercises 5 and 6, determine whether the statement is true or false. If it is false, rewrite it as a true statement.

When the test statistic for the chi-square independence test is large, you will, in most cases, reject the null hypothesis.

In Exercises 5–20, conduct the hypothesis test and provide the test statistic and the P-value and/or critical value, and state the conclusion.

Flat Tire and Missed Class A classic story involves four carpooling students who missed a test and gave as an excuse a flat tire. On the makeup test, the instructor asked the students to identify the particular tire that went flat. If they really didn’t have a flat tire, would they be able to identify the same tire? The author asked 41 other students to identify the tire they would select. The results are listed in the following table (except for one student who selected the spare). Use a 0.05 significance level to test the author’s claim that the results fit a uniform distribution. What does the result suggest about the likelihood of four students identifying the same tire when they really didn’t have a flat?

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Exercises 1–5 refer to the sample data in the following table, which summarizes the frequencies of 500 digits randomly generated by Statdisk. Assume that we want to use a 0.05 significance level to test the claim that Statdisk generates the digits in a way that they are equally likely.

Is the hypothesis test left-tailed, right-tailed, or two-tailed?

Cybersecurity The table below lists the frequency of leading digits of Internet traffic interarrival times for a computer, along with the percentages of each leading digit expected with Benford’s law.

a. Identify the general notation used for observed and expected values.

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Cybersecurity The table below lists the frequency of leading digits of Internet traffic interarrival times for a computer, along with the percentages of each leading digit expected with Benford’s law.

c. Use the results from part (b) to find the contribution to the x2 test statistic from the category representing the leading digit of 2.

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In Exercises 5–20, conduct the hypothesis test and provide the test statistic and the P-value and/or critical value, and state the conclusion.

Heights Measured or Reported? A random sample of the last digits of heights (in.) of males from Data Set 4 “Measured and Reported” is summarized in the table below. Use these last digits to determine whether they occur with about the same frequency. Use a 0.05 significance level. Do the corresponding heights appear to be measured or reported?

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In Exercises 5–20, conduct the hypothesis test and provide the test statistic and the P-value and/or critical value, and state the conclusion.

Testing a Slot Machine The author purchased a slot machine (Bally Model 809) and tested it by playing it 1197 times. There are 10 different categories of outcomes, including no win, win jackpot, win with three bells, and so on. When testing the claim that the observed outcomes agree with the expected frequencies, the author obtained a test statistic of x2 = 8.815 Use a 0.05 significance level to test the claim that the actual outcomes agree with the expected frequencies. Does the slot machine appear to be functioning as expected?

In Exercises 5–20, conduct the hypothesis test and provide the test statistic and the P-value and/or critical value, and state the conclusion.

Bias in Clinical Trials? Researchers investigated the issue of race and equality of access to clinical trials. The following table shows the population distribution and the numbers of participants in clinical trials involving lung cancer (based on data from “Participation in Cancer Clinical Trials,” by Murthy, Krumholz, and Gross, Journal of the American Medical Association, Vol. 291, No. 22). Use a 0.01 significance level to test the claim that the distribution of clinical trial participants fits well with the population distribution. Is there a race/ethnic group that appears to be very underrepresented?