Ch7 Chi-Square Tests For Categorical Exam Questions

Statistics - Unlocking the Power of Data, 3e (Lock)

Chapter 7 Chi-Square Tests for Categorical Variables

7.1 Testing Goodness-of-Fit for a Single Categorical Variable

Use the following to answer the questions below:

Are all colors equally likely for Milk Chocolate M&M's? Data collected from a bag of Milk Chocolate M&M's are provided.

1) State the null and alternative hypotheses for testing if the colors are not all equally likely for Milk Chocolate M&M's.

: = = = = = =

: Some pi is not .

Diff: 2 Type: ES Var: 1

L.O.: 7.1.0;7.1.1

2) If all colors are equally likely, how many candies of each color (in a bag of 420 candies) would we expect to see?

Diff: 2 Type: SA Var: 1

L.O.: 7.1.0;7.1.1

3) Is a chi-square test appropriate in this situation?

A) Yes

B) No

Diff: 2 Type: MC Var: 1

L.O.: 7.1.2

4) How many degrees of freedom are there?

A) 2

B) 3

C) 4

D) 5

Diff: 2 Type: BI Var: 1

L.O.: 7.1.0;7.1.1

5) Calculate the chi-square test statistic. Report your answer with three decimal places.

Diff: 2 Type: SA Var: 1

L.O.: 7.1.1

6) Report the p-value for your test. What conclusion can be made about the color distribution for Milk Chocolate M&M's? Use a 5% significance level.

There is very strong evidence that the six colors are not all equally likely among Milk Chocolate M&M's.

Diff: 2 Type: ES Var: 1

L.O.: 7.1.0;7.1.1

7) Which color contributes the most to the chi-square test statistic? For this color, is the observed count smaller or larger than the expected count?

Diff: 2 Type: ES Var: 1

L.O.: 7.1.0

8) Are all colors equally likely for Dark Chocolate M&M's? Data collected from a bag of Dark Chocolate M&M's are provided.

Test, at the 5% level, if this sample provides evidence that not all colors are equally likely for Dark Chocolate M&M's. Include all details of the test.

: Some pi is not .

Test Contribution

Category Observed Proportion Expected to Chi-Sq

Blue 77 0.166667 72.5 0.27931

Brown 74 0.166667 72.5 0.03103

Green 57 0.166667 72.5 3.31379

Orange 81 0.166667 72.5 0.99655

Red 62 0.166667 72.5 1.52069

Yellow 84 0.166667 72.5 1.82414

N DF Chi-Sq P-Value

435 5 7.96552 0.158

The expected counts are all larger than 5, so it is appropriate to perform the chi-square test.

Test statistic: = 7.96552

Degrees of freedom = 5

p-value = 0.158

There is no evidence to reject and thus there is no evidence that the six colors are not equally likely in Dark Chocolate M&M's.

Diff: 2 Type: ES Var: 1

L.O.: 7.1.1

Use the following to answer the questions below:

An insurance agent is interested in knowing if car crashes are more likely to occur on some days of the week than others. She selects a random sample of 250 insurance claims involving car crashes. Computer output from her chi-square test is provided.

9) Is a chi-square test appropriate in this situation?

A) Yes

B) No

Diff: 2 Type: MC Var: 1

L.O.: 7.1.2

10) Test, at the 5% level, if there is evidence that car crashes are not equally like to occur on all days of the week. Include all details of the test.

: Some pi is not .

Test statistic: = 4.632

Degrees of freedom: 6

p-value = 0.592

There is no evidence to reject and thus there is no evidence to conclude that car crashes occur with differing probabilities on the seven days of the week.

Diff: 2 Type: ES Var: 1

L.O.: 7.1.1

11) The contribution for Friday is missing. Compute the contribution for Friday. Report your answer with three decimal places.

Diff: 2 Type: SA Var: 1

L.O.: 7.1.0;7.1.1

Use the following to answer the questions below:

Observed counts from a sample are provided in the following table. The expected counts from a null hypothesis are given in parentheses.

12) What is the - test statistic?

A) 3.941

B) 3.711

C) 4.315

D) 2.983

Diff: 2 Type: BI Var: 1

L.O.: 7.1.0

13) How many degrees of freedom are there?

A) 1

B) 2

C) 3

D) 4

Diff: 2 Type: BI Var: 1

L.O.: 7.1.0

14) Based on the expected counts, which of the following is most likely the null hypothesis?

A) : = = =

B) : = 0.25, = 0.25, = 0.5

C) : = 0.2, = 0.4, = 0.4

D) : = 0.5, = 0.2, = 0.3

Diff: 3 Type: BI Var: 1

L.O.: 7.1.0

Use the following to answer the questions below:

In a survey conducted by the Gallup organization, 1,017 adults were asked "In general, how much trust and confidence do you have in the mass media — such as newspapers, TV, and radio — when it comes to reporting the news fully, accurately, and fairly?" The results are summarized in the provided table.

We are interested in testing whether or not the four responses are equally likely.

15) Is a chi-square test appropriate in this situation?

A) Yes

B) No

Diff: 2 Type: MC Var: 1

L.O.: 7.1.2

16) Test, at the 5% level, if there is evidence that the four opinions are not all equally likely. Include all details of the test.

: = = = =

: some pi is not

Test Contribution

Category Observed Proportion Expected to Chi-Sq

"Great deal" of confidence 81 0.25 254.25 118.055

"Fair amount" of confidence 325 0.25 254.25 19.688

"Not very much" confidence 397 0.25 254.25 80.148

"No confidence at all" 214 0.25 254.25 6.372

N DF Chi-Sq P-Value

1017 3 224.263 0.000

Test Statistic: = 224.263

Degrees of freedom: 3

p-value ≈ 0

There is very strong evidence that the four opinions are not all equally likely.

Diff: 2 Type: ES Var: 1

L.O.: 7.1.1

17) Which opinion has the largest contribution to the chi-square test statistic? For this age group, is the observed count smaller or larger than the expected count?

Diff: 2 Type: ES Var: 1

L.O.: 7.1.0

Use the following to answer the questions below:

Upon request, the Mars Company (the maker of M&M's) will provide the color distribution for their candies. As of August 2009, they noted that

"Our color blends were selected by conducting consumer preference tests, which indicate the assortment of colors that pleased the greatest number of people and created the most attractive overall effect.

On average, our mix of colors for M&M'S CHOCOLATE CANDIES is:

M&M'S MILK CHOCOLATE: 24% cyan blue, 20% orange, 16% green, 14% bright yellow, 13% red, 13% brown."

Data collected from a bag of Milk Chocolate M&M's are provided.

We want to determine if this sample provides evidence that the color distribution has changed since August 2009.

18) State the null and alternative hypotheses for testing if the color distribution for Milk Chocolate M&M's has changed since 2009.

: = 0.24, = 0.13, = 0.16, = 0.20, = 0.13, = 0.14

: One of the equalities in does not hold.

Diff: 1 Type: ES Var: 1

L.O.: 7.1.0;7.1.1

19) Find the expected counts for each color using the sample size (420 total candies) and null hypothesis.

Diff: 2 Type: ES Var: 1

L.O.: 7.1.0;7.1.1

20) Is a chi-square test appropriate in this situation?

A) Yes

B) No

Diff: 2 Type: MC Var: 1

L.O.: 7.1.2

21) How many degrees of freedom are there?

A) 2

B) 3

C) 4

D) 5

Diff: 2 Type: BI Var: 1

L.O.: 7.1.0;7.1.1

22) Report the chi-square test statistic. Use three decimal places.

Diff: 2 Type: SA Var: 1

L.O.: 7.1.0;7.1.1

23) Report the p-value for your test. What conclusion can be made about the color distribution of Milk Chocolate M&M's? Use a 5% significance level.

A) p-value = 0.048

We have evidence that the color distribution of Milk Chocolate M&M's has changed since 2009.

B) p-value = 0.048

We have do not have evidence that the color distribution of Milk Chocolate M&M's has changed since 2009.

C) p-value = 0.052

We have evidence that the color distribution of Milk Chocolate M&M's has changed since 2009.

D) p-value = 0.052

We have do not have evidence that the color distribution of Milk Chocolate M&M's has changed since 2009.

Diff: 2 Type: BI Var: 1

L.O.: 7.1.0;7.1.1

24) Which color contributes the most to the chi-square test statistic? For that color, is the observed count larger or smaller than what we would expect under the null hypothesis?

Diff: 2 Type: ES Var: 1

L.O.: 7.1.0

Use the following to answer the questions below:

The Gallup organization surveyed a random sample of American adults about their belief in the theory of evolution. The responses are summarized in the provided table.

25) Is a chi-square test appropriate for testing if all beliefs are not equally likely?

A) Yes

B) No

Diff: 2 Type: MC Var: 1

L.O.: 7.1.2

26) Test, at the 5% level, if there is evidence that not all opinions are equally likely.

: = = =

: Some pi is not .

Test Contribution

Category Observed Proportion Expected to Chi-Sq

Believe 397 0.333333 339.333 9.7999

Do Not Believe 254 0.333333 339.333 21.4591

No Opinion 367 0.333333 339.333 2.2557

N DF Chi-Sq P-Value

1018 2 33.5147 0.000

Test statistic: = 33.5147

Degrees of freedom: 2

p-value ≈ 0

There is very strong evidence that not all opinions are equally likely.

Diff: 2 Type: ES Var: 1

L.O.: 7.1.1

27) Which opinion contributes the most to the chi-square test statistic? For that opinion, is the observed count larger or smaller than we would expect?

Diff: 2 Type: ES Var: 1

L.O.: 7.1.0

28) In a survey, Gallup asked a random sample of U.S. adults if they would prefer to have a job outside the home, or if they would prefer to stay home to care for the family and home. The results are summarized below.

Use the goodness-of-fit test to determine if there is evidence that the two choices are not equally likely. Use a 5% significance level.

: = =

: The pi's are not the same.

Test Contribution

Category Observed Proportion Expected to Chi-Sq

Job Outside of Home 645 0.5 488.5 50.1377

Stay at Home 332 0.5 488.5 50.1377

N DF Chi-Sq P-Value

977 1 100.275 0.000

Both expected counts are larger than 5, so it is appropriate to use the chi-square test.

Test statistic: = 100.275

Degrees of freedom: 1

p-value ≈ 0

There is very strong evidence that the two choices are not equally popular.

Diff: 2 Type: ES Var: 1

L.O.: 7.1.1

7.2 Testing for an Association between Two Categorical Variables

Use the following to answer the questions below:

February 12, 2009 marked the 200th anniversary of Charles Darwin's birth. To celebrate, Gallup, a national polling organization, surveyed 1,018 Americans about their education level and their beliefs about the theory of evolution. The survey results are displayed in the provided two-way table. Note that the expected counts for most cells appear in parentheses.

1) Compute the expected cell count for the (College Graduate, Do Not Believe) cell. Report your answer with one decimal place.

Diff: 2 Type: SA Var: 1

L.O.: 7.2.0;7.2.1

2) Compute the contribution to the chi-square statistic for the (Postgraduate, Believe) cell. Report your answer to two decimal places.

Diff: 2 Type: SA Var: 1

L.O.: 7.2.0;7.2.1

3) What are the degrees of freedom for the test?

A) 6

B) 4

C) 3

D) 11

Diff: 2 Type: BI Var: 1

L.O.: 7.2.0;7.2.1

4) State the hypotheses for testing whether the data indicate that there is some association between education level and belief in evolution.

: Belief about evolution does not depend on education level.

: Belief about evolution is related to education level.

Diff: 2 Type: ES Var: 1

L.O.: 7.2.0;7.2.1

5) Is it appropriate to use a chi-square test to test for an association between education level and belief about evolution?

A) Yes

B) No

Diff: 2 Type: MC Var: 1

L.O.: 7.2.2

6) Using a 5% significance level and assuming the test statistic is = 127.451, compute the p-value and make an appropriate conclusion for this test. If there is a significant association between these two variables, describe how they are related.

There is very strong evidence of a significant association between education level and belief about evolution. Individuals who are more educated are more likely to believe in the theory of evolution (or have an opinion) than those who are less educated.

Diff: 2 Type: ES Var: 1

L.O.: 7.2.0;7.2.1

7) A study to investigate the dominant paw in cats was described in the scientific journal Animal Behaviour. The researchers used a random sample of 42 domestic cats. In this study, each cat was shown a treat (5 grams of tuna), and while the cat watched, the food was placed inside a jar. The opening of the jar was small enough that the cat could not stick its head inside to remove the treat. The researcher recorded the paw that was first used by the cat to try to retrieve the treat. This was repeated 100 times for each cat (over a span of several days). The paw used most often was deemed the dominant paw.

The researchers want to determine if there is a significant association between sex of the cat and dominant paw. Computer output from the analysis is provided. Is it appropriate to perform the chi-test to test for an association between sex and dominant paw in cats? If so, perform the test. If not, briefly explain why.

Rows: Sex Columns: Paw

Pearson Chi-Square = 30.927, DF = 1, P-Value = 0.000

: Sex and dominant paw in cats are related.

All expected cell counts are larger than 5, so it is appropriate to use the chi-square test.

Test Statistic: 30.927

Degrees of freedom: 1

p-value ≈ 0

There is very strong evidence that sex and the dominant paw in cats are related.

Diff: 2 Type: ES Var: 1

L.O.: 7.2.2

8) M&M's, the popular candy-coated chocolate treats, come in a variety of flavors. One of the newest varieties is Pretzel, and another popular variety is Peanut Butter. Does the Mars Company (the maker of M&M's) use the same color distribution (frequency of colors) for all varieties, or does it depend on variety? Data collected on the two varieties are displayed in the provided two-way table. Test, at the 5% level, if the samples provide evidence of an association between color and variety. Include all of the details of the test.

: Color does not depend on variety.

: Color does depend on variety.

Rows: Variety Columns: Color

Blue Brown Green Orange Red Yellow All

Pretzel 3 28 11 24 15 24 35

25.50 28.42 20.48 20.48 20.48 19.64 135.00

2.2090 0.0062 4.3881 0.6050 1.4663 0.9659 *

Peanut Butter 28 40 38 25 34 23 188

35.50 39.58 28.52 28.52 28.52 27.36 188.00

1.5863 0.0045 3.1510 0.4345 1.0529 0.6936 *

All 61 68 49 49 49 47 323

61.00 68.00 49.00 49.00 49.00 47.00 323.00

* * * * * * *

Cell Contents: Count

Expected count

Contribution to Chi-square

Pearson Chi-Square = 16.563, DF = 5, P-Value = 0.005

All expected cell counts are greater than 5, so it is appropriate to use the chi-square test.

The test statistic is = 16.563. There are 5 degrees of freedom. The p-value is 0.005.

There is very strong evidence to reject and thus there is very strong evidence to conclude the color distribution does depend on variety.

Diff: 2 Type: ES Var: 1

L.O.: 7.2.1

Use the following to answer the questions below:

M&M's, the popular candy-coated chocolate treats, come in a variety of flavors. Two popular varieties are Milk Chocolate (sometimes referred to as "Plain") and Peanut. Does the Mars Company (the maker of M&M's) use the same color distribution (frequency of colors) for all varieties, or does it depend on variety? Data were collected on the two varieties and computer output for a chi-square test of association is provided.

Rows: Variety Columns: Color

Pearson Chi-Square = 6.183, DF = 5, P-Value = 0.289

9) Is it appropriate to use a chi-square test to test for an association between variety and color?

A) Yes

B) No

Diff: 2 Type: MC Var: 1

L.O.: 7.2.2

10) Test, at the 5% level, if there is a significant association between variety and color. Include all details of the test.

: Color does not depend on variety.

: Color depends on variety.

All expected cell counts are larger than 5.

Test statistic: = 6.183

Degrees of freedom: 5

p-value = 0.289

There is no evidence to reject and thus there is no evidence of a significant association between variety (Peanut versus Plain) and color. That is, there is no evidence that the colors appear with different frequencies in Milk Chocolate and Peanut M&M's.

Diff: 2 Type: ES Var: 1

L.O.: 7.2.1

Use the following to answer the questions below:

We have a random sample of 150 students (60 males and 90 females) that includes two variables: Smoke = "yes" or "no" and Gender = "female (F)" or "Male (M)." The two-way table below summarizes the results.

11) Is it appropriate to use a chi-square test to test for an association between gender and smoking status?

A) Yes

B) No

Diff: 2 Type: MC Var: 1

L.O.: 7.2.2

12) Test, at the 10% level, if there is a significant association between gender and smoking status among students at this university. Include all of the details of the test.

: Smoking status does not depend on gender.

: Smoking status and gender are related.

Rows: Gender Columns: Smoke

No Yes All

F 81 9 90

79.20 10.80 90.00

0.04091 0.30000 *

M 51 9 60

52.80 7.20 60.00

0.06136 0.45000 *

All 132 18 150

132.00 18.00 150.00

* * *

Cell Contents: Count

Expected count

Contribution to Chi-square

Pearson Chi-Square = 0.852, DF = 1, P-Value = 0.356

Test statistic: = 0.852

Degrees of freedom: 1

p-value = 0.356

There is no evidence, at the 1% level, of an association between gender and smoking status.

Diff: 2 Type: ES Var: 1

L.O.: 7.2.1

Use the following to answer the questions below:

A political science professor at a small university wants to know if political party affiliation is significantly associated with trust in the media. He randomly selects 66 Democrats, 63 Republicans, and 41 Independents. Computer output of his chi-square analysis is provided.

Rows: Party Columns: Trust Media?

Pearson Chi-Square = 15.320, DF = 2, P-Value = 0.000

13) The expected count for the (Independent, Yes) cell is missing. Compute the expected count for this cell. Report your answer with two decimal places.

Diff: 2 Type: SA Var: 1

L.O.: 7.2.0;7.2.1

14) Is it appropriate to use a chi-square test to test for an association between political party and trust in the media?

A) Yes

B) No

Diff: 2 Type: MC Var: 1

L.O.: 7.2.2

15) Test, at the 5% level, if there is a significant association between political party affiliation and trust in the media. Include all details of the test.

: Political party affiliation and trust in the media are not related.

: Political party affiliation and trust in the media are related.

Test statistic: = 15.32

Degrees of freedom: 2

p-value ≈ 0

There is very strong evidence to reject and thus there is very strong evidence to conclude that there is a significant association between political party affiliation and trust in the media.

Diff: 2 Type: ES Var: 1

L.O.: 7.2.1

16) Which cell has the largest contribution to the chi-square statistic? For this cell, is the observed count larger or smaller than the expected count?

Diff: 2 Type: ES Var: 1

L.O.: 7.2.0

Use the following to answer the questions below:

The Gallup organization recently conducted a survey of 1,015 randomly selected U.S. adults about "Black Friday" shopping. They asked the following question:

"As you know, the Friday after Thanksgiving is one of the biggest shopping days of the year.

Looking ahead, do you personally plan on shopping on the Friday after Thanksgiving, or not?"

Their results, broken down by sex, are summarized in the provided two-way table.

17) Compute the expected cell counts for all cells. Report your counts to two decimal places.

Diff: 2 Type: ES Var: 1

L.O.: 7.2.0;7.2.1

18) Is it appropriate to use a chi-square test to test for an association between sex and plans to shop the Friday after Thanksgiving?

A) Yes

B) No

Diff: 2 Type: MC Var: 1

L.O.: 7.2.2

19) Test, at the 1% level, if there is a significant association between sex and plans to shop the Friday after Thanksgiving. Include all details of your test.

: Plans to shop the Friday after Thanksgiving does not depend on sex.

: Sex and plans to shop the Friday after Thanksgiving are related.

Rows: Sex Columns: Plans to Shop

No Yes All

Female 400 100 500

410.3 89.7 500.0

0.2608 1.1936 *

Male 433 82 515

422.7 92.3 515.0

0.2532 1.1589 *

All 833 182 1015

833.0 182.0 1015.0

* * *

Cell Contents: Count

Expected count

Contribution to Chi-square

Pearson Chi-Square = 2.866, DF = 1, P-Value = 0.090

Test statistic: = 2.866

Degrees of freedom: 1

p-value = 0.09

Because the p-value is smaller than the 10% significance level, we have some (somewhat weak) evidence to reject and thus have some (somewhat weak) evidence that there is a significant association between sex and plans to shop on the Friday after Thanksgiving.

Diff: 2 Type: ES Var: 1

L.O.: 7.2.1

20) The Gallup organization asked a random sample of U.S. adults if they would prefer to have a job outside the home, or if they would prefer to stay home to care for the family and home. Of the 504 males they surveyed, 391 said that they would prefer to have a job outside of the home. Of the 473 females they surveyed, 254 said that they would prefer a job outside of the home.

Test, at the 5% level, if there is evidence of an association between sex and preference to have a job outside of the home. Include all details of the test.

: Job preference does not depend on sex.

: Sex and job preference are related.

Rows: Sex Columns: Job Preference

Outside Stay at

Home Home All

Female 254 219 473

312.3 160.7 473.0

10.87 21.12 *

Male 391 113 504

332.7 171.3 504.0

10.20 19.82 *

All 645 332 977

645.0 332.0 977.0

* * *

Cell Contents: Count

Expected count

Contribution to Chi-square

Pearson Chi-Square = 62.021, DF = 1, P-Value = 0.000

Test statistic: = 62.021

Degrees of freedom: 1

p-value = 0

There is very strong evidence to reject and thus there is very strong evidence of a significant association between job preference and sex.

Diff: 2 Type: ES Var: 1

L.O.: 7.2.1

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