AP®StatisticsCollege BoardExam QuestionsUnits 6-9: InferenceIndependence & Homogeneity (Chi-Square)

Independence & Homogeneity (Chi-Square) (College Board AP® Statistics)

Exam Questions

5 mins5 questions
1
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An online coffee bean retailer is investigating whether the type of coffee beans that their customers buy and the age of the customers are related. Each customer in a random sample of size 330 from their customer list is categorized by their age and the type of coffee beans they order. The data are in the table below.

Coffee bean type

Arabica

Robusta

Liberica

Excelsa

Age (years)

Less than 30

48

61

57

17

Greater than or equal to 30

29

45

50

23

Which of the following procedures would be most appropriate to use for investigating whether there is a relationship between age and coffee bean type?

  • A two-sample z-test

  • A two-sample t-test

  • A chi-square test for homogeneity

  • A chi-square goodness of fit test

  • A chi-square test for independence

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2
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A dog food company performed an experiment with a large sample of Dalmatians, bulldogs, and poodles. Each dog was presented with a number of bowls, each containing a different type of dog food. The first bowl contained "Yum-dog," the second "Woofy," and the third "Meatalicious." It was assumed that the first type of dog food that each dog ate was their preferred type of dog food.

A chi-square test is to be used to determine whether the preferred type of dog food is independent of the dog breed. The number of degrees of freedom for the chi-square test in this situation is

  • 9

  • 6

  • 4

  • 3

  • 1

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3
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A random sample of 144 people, made up of adults and children, was asked to select their preferred sport to watch in the Olympic Games from the options of gymnastics, swimming, and running. The number of adults and children who selected each sport is shown in the table below.

Age

Adults

Children

Preferred sport

Gymnastics

18

32

Swimming

23

21

Running

21

29

In a test of independence of age and preferred sport to watch, which of the following is used as the expected cell count for adults who prefer to watch swimming?

  • fraction numerator open parentheses 23 plus 21 close parentheses open parentheses 18 plus 23 plus 21 close parentheses over denominator 144 end fraction

  • 1 third open parentheses 18 plus 23 plus 21 close parentheses

  • 1 half open parentheses 23 plus 21 close parentheses

  • 1 over 6 open parentheses 144 close parentheses

  • 23

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4
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A randomly selected sample of 100 workers from various cities across a state was surveyed about the type of transport they used to commute to work. The cities in which they worked were categorized as "small" or "large," and the transport options included walking, public transport, car, or bicycle. The data are summarized in the table below.

Size of city

Small

Large

Type of transport

Walking

13

11

Public transport

21

23

Car

11

9

Bicycle

8

4

In testing to see whether the size of the city worked in is independent of the type of transport used to commute to work, a chi-square statistic of 1.44 and a p-value more than 0.6 were calculated. Which of the following statements is correct?

  • The chi-square test should not have been used because the value for people who commute to work by bicycle in a large city is less than 5.

  • The null hypothesis states that there is an association between the size of the city and the type of transport used to commute to work, and the large p-value suggests that the null hypothesis should not be rejected.

  • The number of degrees of freedom for the test is 8.

  • The chi-square test shows that people from small cities favor public transport to commute to work.

  • The large p-value suggests that there is not sufficient evidence of an association between the size of the city and the type of transport used to commute to work.

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5
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An advertising executive hypothesizes that parents in California and parents in Arizona differ in their preferred brand of infant formula milk. The advertising executive collects data from 200 randomly selected parents in California and 170 randomly selected parents in Arizona on their preferred brand of four types of infant formula. A chi-square test for homogeneity of proportions will be used to test the hypothesis. Assuming that the conditions for inference are met, which of the following statements is true for the test?

  • The null hypothesis for the test is that the proportion of parents from each state who prefer each brand of formula is 1 fourth.

  • The test would also be appropriate if 100 children from each state had been used for the two samples.

  • The sample size is too small to detect a significant difference between infant formula preferences for parents in different states.

  • The test is not valid because the sample sizes are unequal.

  • The more the parents from California and Arizona differ in their infant formula preferences, the smaller the p-value will be.

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