Lesson 11.2.1 Introduction to Chi-Square Tests for Two-Way ......

© 2011 THE CARNEGIE FOUNDATION FOR THE ADVANCEMENT OF TEACHING A PATHWAY THROUGH STATISTICS, VERSION 1.5, STATWAY™ - STUDENT HANDOUT

STATWAY™ STUDENT HANDOUT

Lesson 11.2.1 Introduction to Chi-Square Tests for Two-Way Tables

STUDENT NAME DATE

INTRODUCTION

A survey is conducted of 1,000 randomly selected moviegoers who just saw a new, highly anticipated

science fiction/action film. In the survey, the moviegoers were asked if they liked the film (yes or no) and if

they considered themselves very knowledgeable about science fiction, moderately knowledgeable about

science fiction, or having little to no knowledge about science fiction. It is important for the company that

made the movie to know if the movie was more popular among certain groups. This knowledge might affect

the movie’s future advertising strategy and future marketing campaign.

The company is interested in determining if the moviegoer’s opinion of the film is dependent on the

moviegoer’s self reported science fiction knowledge level. Recall from Module 6, that two variables are not

independent if the presence of one variable influences the presence of the other.

TRY THESE

1 A table such as the one below might appear in a report summarizing the results of the survey.

Did you like the film?

Yes No TOTAL

Very Knowledgeable 400

Moderately Knowledgeable 250

Having Little to No Knowledge 350

TOTAL 300 700 1000

Response

Self-reported Science Fiction

Knowledge Level

STATWAY STUDENT HANDOUT | 2



In the table, notice that a proportion of 0.30 of the entire group liked the film (300/1,000).

Assuming that the variables Self-reported Science Fiction Knowledge Level and Response are

independent, answer the following questions.

A How many of the 400 Very Knowledgeable moviegoers would you expect to answer Yes, and

how many would you expect to answer No? Fill in the appropriate two cells with your answers.

B How many of the 250 Moderately Knowledgeable moviegoers would you expect to answer,

“Yes,” and how many would you expect to answer “No”? Fill in the appropriate two cells with your answers.

C How many of the 350 “Having Little to No Knowledge” moviegoers would you expect to

answer Yes, and how many would you expect to answer No? Fill in the appropriate cells with your answers.

D Verify that the values you just added to the table support (add up to) the appropriate row

and column totals shown in the margins of the table.

Based on sampling variability, you know that it would be unusual to obtain a different sample with the exact

same counts as the table in Question 1 even if the row and columns variables were truly independent for the

entire population.




2 For another sample of 1,000 individuals fill in the following table below in such a way that the data

would not lead you to challenge the claim that the variables Response and Self-reported Science

Fiction Knowledge Level are independent. Make sure your entries add up to the correct row and

column totals.


3 For another sample of 1,000 individuals, fill in the following table in such a way that the data

would provide strong evidence against the claim that the variables Response and Self-reported

Science Fiction Knowledge Level are independent. Make sure your entries add up to the correct

row and column totals.


Yes No TOTAL




TOTAL 300 700 1000

Response


Knowledge Level

Yes No TOTAL




TOTAL 300 700 1000

Response


Knowledge Level




4 Below are the distributions of three hypothetical random samples of size 1,000. For each sample,

examine the proportions of Yes responses for each category of Self-reported Science Fiction

Knowledge Level and determine whether you think the sample provides strong evidence,

moderate evidence, or weak evidence against the claim that the variables Response and Self-

reported Science Fiction Knowledge Level are independent. Then list the characteristics of the

sample that led you to your decision.

Hypothetical Sample 1: “Did you like the film?”

It appears that Sample 1 provides (circle one)

strong moderate weak

evidence against the claim that the variables Response and Self-reported Science Fiction

Knowledge Level are independent because…


It appears that Sample2 provides (circle one)




Yes No TOTAL

Very Knowledgeable 300 100 400

Moderately Knowledgeable 200 50 250

Having Little to No Knowledge 100 250 350

TOTAL 600 400 1000

Response


Knowledge Level

Yes No TOTAL




TOTAL 462 538 1000

Response


Knowledge Level





It appears that Sample3 provides (circle one)




5 For each hypothetical sample, develop a graphical display that visually compares the proportions

of Yes responses for each category of Self-reported Science Fiction Knowledge Level. For each

sample’s graph, state the most notable feature of the graph. Does the information shown in each

graph correspond in any way to your initial “strength of evidence” statements that you made

above in Question 4?

Sample 1 graphical display:

Most notable feature:

Yes No TOTAL




TOTAL 450 550 1000

Response


Knowledge Level








6 Based on your work in the previous questions, which sample gives the strongest evidence against

the claim that the variables Response and Self-reported Science Fiction Knowledge Level are

independent? What characteristics of the sample were most important in your decision?

7 It would be useful to have a statistical measure of deviation to determine how much the

distribution of a sample (such as those shown previously) deviates from what is expected. Create a

method (or a statistic) to measure which sample deviates the most from its ideal expected

distribution. How is your method similar to methods discussed in previous lessons? How is it

different?




NEXT STEPS Quantifying the Strength of the Evidence Revisiting Independence in the Context of Expected Counts In Question 1, you filled in a table under the assumption that the variables Self-reported Science Fiction

Knowledge Level and Response were independent. To fill in the count for the number of Very

Knowledgeable moviegoers who said Yes, it was important to account for the fact that that 0.30 of the

entire survey said Yes and that 400 individuals (i.e., 0.40 of the entire survey) responded as being Very

Knowledgeable.

Since 0.30 of the entire survey said Yes, under the assumption of independence, the conditional

frequency distribution for each category of Self-reported Science Fiction Knowledge Level was 0.30 Yes

and 0.70 No.

This means that the number of Yes responses for each category of Self-reported Science Fiction

Knowledge Level was:

0.30 of the Very Knowledgeable group = 0.30 × 400 = 120

0.30 of the Moderately Knowledgeable group = 0.30 × 250 = 75

0.30 of the Having Little to No Knowledge group = 0.30 × 350 = 105

The expected count of 120 for the number of moviegoers who were Very Knowledgeable and said Yes is

directly related to three pieces of information: 0.40 of the entire survey responded as being Very

Knowledgeable, 0.30 of the entire survey said Yes, and 1,000 people were surveyed.

Said another way, the expected count for Very Knowledgeable and Yes equals the proportion of those

surveyed who said Very Knowledgeable × proportion of those surveyed who said Yes × total number

surveyed = 0.40 ×0.30 × 1,000 = 120.

Generally speaking, for a two-way table, under the assumption of independence, any expected count for a

cell that represents the combination of a row variable category with a column variable category can be

computed based on the cell’s corresponding row proportion, corresponding column proportion, and the

total number of observations in the table (called the grand total).




As a formula this is expressed as follows.

Through algebra, there is another form of the above formula that is based on raw counts from the table:

Each proportion is equal to the total for each category divided by the grand total.

Simplifying the above expression yields the final form.

8 For the table in Question 1, you filled in the six counts under the assumption that

the variables Self-reported Science Fiction Knowledge Level and Response were independent.

Thus, the values you filled in should have been the appropriate expected counts for each

combination of a row variable category and a column variable category if the claim of

independence between the row and column variables were true for the population that the 1,000

randomly selected moviegoers represent. Verify that the counts you developed based on the

respective row and column totals follow the formula above (or change your expected counts as

needed to adhere to the formula and to the row and column totals in the table).

Yes No TOTAL

Very Knowledgeable 400 * 300/1000

= 120

400 * 700/1000

= 280400

Moderately Knowledgeable 250 * 300/1000

= 75

250 * 700/1000

= 175250

Having Little to No Knowledge350 * 300/1000

= 105

350 * 700/1000

= 245350

TOTAL 300 700 1000

Response

Self-reported

Science Fiction

Knowledge Level




Computing a Chi-Square Statistic For two-way tables, you can compute a chi-square statistic that is useful in assessing the strength of your

evidence against the claim of independence. The mechanics of computing the chi-square statistic for a two-

way table are very similar to the methods introduced in a previous lesson for developing the chi-square

statistic for a one-way table. The differences are as follows:

Expected counts are computed for each combination of a row variable category with a column

variable category based on the corresponding row totals and column totals. (Note: The Total

row and Total column presented in a table do not represent a category of the variable of

interest. Any presentation of a row or column total in the table is not considered as a category

of a given variable.)

Computations comparing observed counts with expected counts are performed for each

combination of a row variable category with a column variable category.

The expected count for a combination of a row variable category with a column variable

category is computed as shown previously:

As before, when calculating expected counts, it is okay if the expected counts are non integer values and you

should generally not round expected counts to whole numbers. Also note that the sum of your expected

counts for a given row should equal that row’s total in your sample and the sum of your expected counts for

a given column should equal that column’s total in your sample.

The expected counts for Hypothetical Sample 1 would be as follows:


Yes No TOTAL

Very Knowledgeable 400 * 600/1000

= 240

400 * 400/1000

= 160400

Moderately Knowledgeable 250 * 600/1000

= 150

250 * 400/1000

= 100250

Having Little to No Knowledge350 * 600/1000

= 210

350 * 400/1000

= 140350

TOTAL 600 400 1000

Response

Self-reported

Science Fiction

Knowledge Level




Once you have computed expected counts for each row and column variable combination, the steps for

computing a chi-square value for a two-way table are as follows:

(1) For each combination of row variable category and column variable category, compute the

difference between the actual count for that combination (obtained from the sample) and the expected count for that combination:

(Observed Count – Expected Count)

(2) For each combination of row variable category and column variable category, compute the

square of the difference obtained in Step 1:

(Observed Count – Expected Count)2

(3) For each combination of row variable category and column variable category, divide the

squared difference obtained in Step 2 by the expected count for the combination:

(Observed Count – Expected Count)2/Expected Count (4) Add up the Step 3 calculation results from each combination; this will be the chi-square

value.

Example: Hypothetical Sample 1

Step 4: 15 + 22.5 + 16.667 + 25 + 57.619 + 86.429 = 223.215

"Yes"

AND "V

ery K

nowle

dgeable

"

"No" A

ND "Very

Know

ledge

able"

"Yes"

AND "M

oderate

ly K

nowle

dgeable

"

"No" A

ND "Modera

tely

Know

ledge

able

"

"Yes"

AND "L

ittle

to N

o Know

ledge

"

"No" A

ND "Litt

le to

No K

nowle

dge"

Observed Count (from sample) 300 100 200 50 100 250

Expected Count (based on claim of independence) 240 160 150 100 210 140

Step 1: Observed Count - Expected Count 60 -60 50 -50 -110 110

Step 2: (Observed Count - Expected Count)23600 3600 2500 2500 12100 12100

Step 3: (Observed Count - Expected Count)2/Expected Count 15 22.5 16.667 25 57.619 86.429




For Hypothetical Sample 1, the chi-square value generated is 223.215.

9 Compute the chi-square values for the other two hypothetical samples by filling in the following

tables. Compute the Step 3 calculations to three decimal places as shown in the example above.

Hypothetical Sample 2

Step 4: 0.997 + + + 0.062 + + 0.154 = 2.322

For Hypothetical Sample 2, the chi-square value generated is 2.322.

Hypothetical Sample 3

Step 4: 90 + + + 14.957 + 89.525 + 73.248 =

"Yes"

AND "V

ery K

nowle

dgeable

"

"No" A

ND "Very

Know

ledge

able"

"Yes"

AND "M

oderate

ly K

nowle

dgeable

"

"No" A

ND "Modera

tely

Know

ledge

able

"

"Yes"

AND "L

ittle

to N

o Know

ledge

"

"No" A

ND "Litt

le to

No K

nowle

dge"


Expected Count (based on claim of independence) 92.4 107.6 143.22 166.78 226.38 263.62

Step 1: Observed Count - Expected Count 9.6 -9.6 -3.22 3.22 -6.38 6.38

Step 2: (Observed Count - Expected Count)292.16 92.16 10.3684 10.3684 40.7044 40.7044

Step 3: (Observed Count - Expected Count)2/Expected Count 0.997 0.857 0.072 0.062 0.180 0.154

"Yes"

AND "V

ery K

nowle

dgeable

"

"No" A

ND "Very

Know

ledge

able"

"Yes"

AND "M

oderate

ly K

nowle

dgeable

"

"No" A

ND "Modera

tely

Know

ledge

able

"

"Yes"

AND "L

ittle

to N

o Know

ledge

"

"No" A

ND "Litt

le to

No K

nowle

dge"


Expected Count (based on claim of independence) 90 110 139.5 170.5 220.5 269.5

Step 1: Observed Count - Expected Count 90 -90 50.5 -50.5 -140.5 140.5

Step 2: (Observed Count - Expected Count)28100 8100 2550.25 2550.25 19740.25 19740.25

Step 3: (Observed Count - Expected Count)2/Expected Count 90 73.6 18.281 14.957 89.525 73.248




For Hypothetical Sample 3, the chi-square value generated is __________.

10 Which of the hypothetical samples had the highest chi-square value? Does the size of the chi-

square values generated by these three samples correspond in any way to the initial strength of

evidence statements that you made in Question 4 or the conjectures you made in Question 6?




TAKE IT HOME 1 Based on your work today, which seems to provide greater evidence against the claim of

independence: a high chi-square or a low chi-square value?

2 A fourth hypothetical sample is as follows:


A Develop a graphical display that visually compares the proportions of Yes responses for each category of Self-reported Science Fiction Knowledge Level.

B Compute the chi-square value and determine if this value provides strong evidence against

the claim that the variables Question Response and Self-reported Science Fiction Knowledge Level are independent.

(Note: For reasons that will be explained in a future lesson, in a two-way table case such as this where one

categorical variable contains three categories and the other categorical variable contains two categories,

consider a chi-square value of 5.99 or greater to be statistically significant evidence against the claim that

the two variables are independent.)

Yes No TOTAL




TOTAL 600 400 1000

Response


Knowledge Level




+++++ This lesson is part of STATWAY™, A Pathway Through College Statistics, which is a product of a Carnegie Networked Improvement Community that seeks to advance student success. Version 1.0, A Pathway Through Statistics, Statway™ was created by the Charles A. Dana Center at the University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching. This version 1.5 and all subsequent versions, result from the continuous improvement efforts of the Carnegie Networked Improvement Community. The network brings together community college faculty and staff, designers, researchers and developers. It is an open-resource research and development community that seeks to harvest the wisdom of its diverse participants in systematic and disciplined inquiries to improve developmental mathematics instruction. For more information on the Statway Networked Improvement Community, please visit carnegiefoundation.org. For the most recent version of instructional materials, visit Statway.org/kernel.

+++++ STATWAY™ and the Carnegie Foundation logo are trademarks of the Carnegie Foundation for the Advancement of Teaching. A Pathway Through College Statistics may be used as provided in the CC BY license, but neither the Statway trademark nor the Carnegie Foundation logo may be used without the prior written consent of the Carnegie Foundation.

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Lesson 11.2.1 Introduction to Chi-Square Tests for Two-Way ......

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