Questions and Answers

A local restaurant claims that it gets 45 percent of its customers from Monday through Thursday, 20 percent on Friday, 20 percent on Saturday, and 15 percent on Sunday. How many degrees of freedom should be used to conduct a chi-square goodness-of-fit test of the claim?
A) 3
B) 4
C) 6
D) 7
E) It is not possible to determine the degrees of freedom without knowing the sample size.

The district manager of four different restaurants wanted to investigate whether the four restaurants differed with respect to customers ordering dessert or not based on family classification (with children or without children). Independent random samples of 100 customers who ordered dessert were selected from each restaurant, and the customers were identified as either being with children or without children. After verifying the conditions for the appropriate hypothesis test, the manager calculated a chi-square test statistic of 6.45 with an associated p-value of 0.092. Based on the p-value and α=0.05, what conclusion should the manager make regarding the proportion of customers who order dessert at each restaurant and the customers’ family classification?
A) There is convincing statistical evidence to suggest that the proportion of customers who order dessert at each restaurant is the same based on family classification.
B) There is convincing statistical evidence to suggest that the proportion of customers who order dessert at each restaurant is not the same based on family classification.
C) There is not convincing statistical evidence to prove that the proportion of customers who order dessert at each restaurant is not the same based on family classification.
D) There is not convincing statistical evidence to suggest that the proportion of customers who order dessert at each restaurant is the same based on family classification.
E) There is not convincing statistical evidence to suggest that the proportion of customers who order dessert at each restaurant is not the same based on family classification.

Polling organizations regularly collect data on the public’s opinions and habits. A question on a recent survey asked, “How often do you purchase coffee from a coffeehouse?” There may be differences in how people respond to this question based on whether the person is a full-time student. Suppose a polling organization uses random digit dialing of local phone numbers to take a poll and asks respondents whether they are full-time students. In addition, they ask respondents to identify how often they purchase coffee from a coffeehouse (never, once a week, two to three times a week, daily, more than once a day). The data are collected in a 5-by-2 table of counts. Which of the following is the appropriate null hypothesis when conducting a chi-square test for the data?
A) H0 : The proportion of people for each category in the 5-by-2 table will be 10 percent.
B) H0 : For each purchasing frequency, the proportion of full-time students is different.
C) H0 : How often a person purchases coffee from a coffeehouse is not associated with whether a person is a full-time student.
D) H0 : How often a person purchases coffee from a coffeehouse is dependent on whether a person is a full-time student.
E) H0 : There is an association between how often a person purchases coffee from a coffeehouse and whether a person is a full-time student.

Which statement is true about whether the conditions for the chi-square test for independence have been met?
A) All necessary conditions are satisfied to apply a chi-square test for independence between gender and being a coffee drinker.
B) The data is not the result of two independent random samples; therefore, the conditions for applying the chi-square test for independence between gender and being a coffee drinker are not met.
C) Not all of the expected cell counts are large enough to satisfy the conditions for applying the chi-square test for independence between gender and being a coffee drinker.
D) Not all of the observed cell counts are large enough to satisfy the conditions for applying the chi-square test for independence between gender and being a coffee drinker.
E) The total sample size is not at least 10 percent of the population of United States adults; therefore, the conditions for applying the chi-square test for independence between gender and being a coffee drinker are not met.

Which of the following is an appropriate description of the chi-square distribution?
A) A chi-square distribution will only contain positive values and will be skewed right, with the skew becoming less pronounced with increasing degrees of freedom.
B) A chi-square distribution will only contain positive values and will be skewed left, with the skew becoming less pronounced with increasing degrees of freedom.
C) A chi-square distribution will only contain positive values and will be skewed right, with the skew becoming more pronounced with increasing degrees of freedom.
D) A chi-square distribution will contain positive and negative values, and will be skewed right, with the skew becoming less pronounced with increasing degrees of freedom.
E) A chi-square distribution will contain positive and negative values and will be skewed left, with the skew becoming less pronounced with increasing degrees of freedom.

A random sample of 500 people were classified by their ages into 3 age-groups: 29 years and younger, 30 to 64 years, and 65 years and older. Each person from the sample was surveyed about which of 4 major brands of cell phone they used. Their responses were compiled and displayed in a 3-by-4 contingency table. A researcher will use the data to investigate whether there is an association between cell phone brand and age-group. Which of the following is the appropriate test for the investigation?
A) A one-sample t-test for a population mean
B) A two-sample t-test for a difference between means
C) A chi-square goodness-of-fit test
D) A chi-square test of homogeneity
E) A chi-square test of independence

A company claims they produce their mixed bag of candies so that, of the candies in the bag, 20 percent are dark chocolate, 60 percent are milk chocolate, and 20 percent are white chocolate. In a random sample of candies of size 50, the counts are as follows: 6 dark, 32 milk, and 12 white. Assuming the conditions for inference are met, what is the test statistic for a chi-square goodness-of-fit test to investigate whether the distribution of the sample is consistent with the company’s claim?
A) χ2=62+322+122
B) χ2=102+302+102
C) χ2=(6−10)2+(32−30)2+(12−10)2
D) χ2=(6−10)210+(32−30)230+(12−10)210
E) χ2=(10−6)26+(30−32)232+(10−12)212

Students in a high school statistics class wanted to see if the distribution of the colors of a popular candy was different in the bags for different types of candies the company manufactures. The students purchased several large bags of regular candies, tropical-flavored candies, and sour-flavored candies. For each type of candy, the students took a random sample of 100 candies and recorded how many of each color (red, green, yellow, or blue) were in the sample. The students verified the conditions for inference and calculated a chi-square test statistic of 12.59 with a corresponding p-value of 0.05. Which of the following is the correct interpretation of the p-value in the context of the test?
A) The hypothesis test has a significance level of α=0.05.
B) There is a 5 percent chance that the distribution of colors is different for the different types of candies.
C) There is a 5 percent chance that the distribution of colors is the same for the different types of candies.
D) Assuming that the distribution of colors for the different types of candies is the same, there is a 5 percent chance of finding a test statistic of 12.59 or larger.
E) Assuming that the distribution of colors for the different types of candies is different, there is a 5 percent chance of finding a test statistic of 12.59 or larger.

How are the expected counts calculated when a chi-square goodness-of-fit test is conducted?
A) The expected counts are calculated by multiplying each proportion in the null hypothesis by 100.
B) The expected counts are calculated by multiplying each proportion in the alternative hypothesis by 100.
C) The values observed from the sample are the expected counts.
D) The expected counts are calculated by multiplying each proportion in the null hypothesis by the sample size.
E) The expected counts are calculated by multiplying each proportion in the alternative hypothesis by the sample size.