13. Chi-Square Tests & Goodness of Fit

"In Problems 7–10, determine (d) test the hypothesis at the level of significance.

H0: pA=pB=pC=pD=pE=1/5

H1: At least one of the proportions is different from the others.

Benford’s Law, Part I Our number system consists of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. The first significant digit in any number must be 1, 2, 3, 4, 5, 6, 7, 8, or 9 because we do not write numbers such as 12 as 012. Although we may think that each first digit appears with equal frequency so that each digit has a 1/9 probability of being the first significant digit, this is not true. In 1881, Simon Newcomb discovered that first digits do not occur with equal frequency. This same result was discovered again in 1938 by physicist Frank Benford. After studying much data, he was able to assign probabilities of occurrence to the first digit in a number as shown.

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Source: T. P. Hill, “The First Digit Phenomenon,” American Scientist, July—August, 1998.

The probability distribution is now known as Benford’s Law and plays a major role in identifying fraudulent data on tax returns and accounting books. For example, the following distribution represents the first digits in 200 allegedly fraudulent checks written to a bogus company by an employee attempting to embezzle funds from his employer.

a. Because these data are meant to prove that someone is guilty of fraud, what would be an appropriate level of significance when performing a goodness-of-fit test?

[DATA] Putting It Together: The V-2 Rocket in London In Thomas Pynchon’s book Gravity Rainbow, the characters discuss whether the Poisson probabilistic model can be used to describe the locations that Germany’s feared V-2 rocket would land in. They divided London into 0.25-km2 regions. They then counted the number of rockets that landed in each region, with the following results:

b. Explain why the requirements for conducting a goodness-of-fit test are not satisfied.

Explain why chi-square goodness-of-fit tests are always right tailed.

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.