13. Chi-Square Tests & Goodness of Fit

Which of the following is not a characteristic of the $\chi ^2$ distribution?

In Exercises 9 and 10, (a) identify the claim and state Ho and Ha , (b) find the critical value(s) and identify the rejection region(s), (c) find the standardized test statistic z, (d) decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim. Assume the samples are random and independent, and the populations are normally distributed.

A researcher claims that the mean sodium content of sandwiches at Restaurant A is less than the mean sodium content of sandwiches at Restaurant B. The mean sodium content of 22 randomly selected sandwiches at Restaurant A is 670 milligrams. Assume the population standard deviation is 20 milligrams. The mean sodium content of 28 randomly selected sandwiches at Restaurant B is 690 milligrams. Assume the population standard deviation is 30 milligrams. At α=0.05, is there enough evidence to support the claim?

Which of the following is an appropriate description of the $\chi -\mathrm{square}$ distribution?

Performing a Chi-Square Goodness-of-Fit Test

In Exercises 7–16, (d) decide whether to reject or fail to reject the null hypothesis.

Ways to Pay A financial analyst claims that the distribution of people’s preferences on how to pay for goods is different from the distribution shown in the figure. You randomly select 600 people and record their preferences on how to pay for goods. The table shows the results. At α=0.01, test the financial analyst’s claim. (Adapted from Travis Credit Union)

Finding Expected Frequencies

In Exercises 3–6, find the expected frequency for the values of n and pᵢ.

n=500, pᵢ=0.9

Which of the following best describes the purpose of the $\chi ²$ goodness of fit test in statistics?

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.

True or False: The shape of the chi-square distribution depends on the degrees of freedom.

Performing a Chi-Square Goodness-of-Fit Test

In Exercises 7–16, (a) identify the claim and state H₀ and Hₐ, (b) find the critical value and identify the rejection region, (c) find the chi-square test statistic, (d) decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim.

Coffee A researcher claims that the numbers of cups of coffee U.S. adults drink per day are distributed as shown in the figure. You randomly select 1600 U.S. adults and ask them how many cups of coffee they drink per day. The table shows the results. At α=0.05, test the researcher’s claim. (Adapted from Gallup)

Performing a Chi-Square Goodness-of-Fit Test

In Exercises 7–16, (b) find the critical value and identify the rejection region.

Ways to Pay A financial analyst claims that the distribution of people’s preferences on how to pay for goods is different from the distribution shown in the figure. You randomly select 600 people and record their preferences on how to pay for goods. The table shows the results. At α=0.01, test the financial analyst’s claim. (Adapted from Travis Credit Union)

Always Wear a Helmet The National Highway Traffic Safety Administration publishes reports about motorcycle fatalities and helmet use. The distribution shows the proportion of fatalities by location of injury for motorcycle accidents.

The following data show the location of injury and number of fatalities for 2068 riders not wearing a helmet.

b. Compare the observed and expected counts for each category. What does this information tell you?

A gym owner wants to know if the gym has similar numbers of members across different age groups. The table shows the distribution of ages for members from a random survey. Write the null & alt. hypotheses to test the claim that the gym has equal numbers of members across all age groups.

Which of the following best describes the null hypothesis of a $\chi 2^{ }$ goodness of fit test?

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.

Which of the following is not a requirement to conduct a $\chi ²$ goodness-of-fit test?