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Chris Morgan, MATH G160 csmorgan@purdue.edu April 13, 2012 Lecture 30. Chapter 2.4: Chi-Squared ( χ 2 ) Test and Independence between two Categorical Variables. Two-Way Tables.

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chris morgan math g160 csmorgan@purdue edu april 13 2012 lecture 30
Chris Morgan, MATH G160

csmorgan@purdue.edu

April 13, 2012

Lecture 30

Chapter 2.4:

Chi-Squared (χ2) Test and Independence between two Categorical Variables

two way tables
Two-Way Tables
  • Any table which allows you to observe multiple pieces of information to help find conditional, joint, and marginal probabilities
  • Expected Counts: the expected count in any cell of a two-way table when the null hypothesis is true
  • The null hypothesis is what you think to be true given previous research, outside readings, or personal opinion based on an educated guess
example 1a
Example 1a:
  • Above is a sample of students in the College of Business. They were asked their chosen major and their sex.
    • What is the probability that a student is a Finance Major?
    • What is the probability that a student is Female?
example 1b
Example 1b:
  • Above is a sample of students in the College of Business. They were asked their chosen major and their sex.
    • 3. What is the probability that a student is female given that the person is in Administration?
example 1c
Example 1c:
  • Above is a sample of students in the College of Business. They were asked their chosen major and their sex.
    • 4. What is the probability that a student is an Administration major given that the student is female?
hypothesis testing
Hypothesis Testing
  • Our null hypothesis is what we expect to see given no interaction between variables
  • Our alternative hypothesis is some improvement or change on the null hypothesis
    • Never accept the Ha
    • Always “reject the Ho” or “fail to reject the Ho”
    • Why?
  • For the chi-square test:
    • Ho: there is no association between two categorical variables, and we conclude they’re independent
    • Ha: there is an association between two categorical variables, and we conclude there is a relationship
calculating a chi squared statistic
Calculating a Chi-Squared Statistic
  • Denoted χ2
  • The observed count is whatever value we see in the table
  • The expected count for each cell in the table can be found by taking:

Note: We can safely use the χ² test under two important conditions:

1. when no more than 20% of the expected counts are less than five

2. when all individual expected counts are one or greater

interpreting a chi squared test
Interpreting a Chi-Squared Test
  • I can compare the calculated chi-square test-statistic to a critical value to see if my variables do in fact have a relationship
  • We will denote the test statistic as χ²* and the critical value as χ²α, (r-1)(c-1) where r is the number of rows, c is the number of columns, and the degrees of freedom is found by: df = (r-1)*(c-1). I can then look up the critical value in the table (see next slide) using the alpha level and df
  • If: | χ²*| > χ²α, (r-1)(c-1)

…then we will reject the null hypothesis and conclude the alternative hypothesis, that the observed values were sufficiently far away from the expected value, meaning it is a significant result and there exists a relationship between the two variables

  • If: | χ²*| ≤ χ²α, (r-1)(c-1)

…then we fail to reject the null hypothesis and the two variables are independent (meaning no relationship exists)

slide10
Chi-Square (χ²) Distribution Critical Values

The first row is the alpha level

The first column is the number of df

example 2a
Example 2a:
  • Returning to example one, is there a relationship between gender and major?
  • Find expected counts
  • Compare expected counts to observed counts
  • Calculate χ²
  • Compare chi-squaretest statistic (χ²*) to chi-square critical value (χ²α, (r-1)(c-1) )
example 2b fill in expected counts
Example 2b: Fill in expected counts

Recall the equation for expected counts:

example 2c calculate
Example 2c: Calculate χ²

Recall the equation for chi-square:

example 2d calculate
Example 2d: Calculate χ²

Recall the equation for chi-square:

Now we just have to add them all together:

and compare the chi-square value to the critical value…

example 2e is significant
Example 2e: Is χ² significant?

To compare the chi-square value to the critical value I look up in the table the value for the chi-squared critical value when alpha = 0.05 and df = 3:

Therefore, since the absolute value of the test statistic is less than or equal to the critical value we (circle one):

reject the Ho fail to reject the Ho accept the Ho accept the Ha

And conclude….what?:

example 3a
Example 3a:
  • Is there a relationship between favorite soda and favorite ice cream?
  • Find expected counts
  • Compare expected counts to observed counts
  • Calculate χ²
  • Compare chi-squaretest statistic (χ²*) to chi-square critical value (χ²α, (r-1)(c-1) )
example 3b fill in expected counts
Example 3b: Fill in expected counts

Recall the equation for expected counts:

example 3c calculate
Example 3c: Calculate χ²

Recall the equation for chi-square:

example 3d calculate
Example 3d: Calculate χ²

Recall the equation for chi-square:

Now we just have to add them all together:

and compare the chi-square value to the critical value…

example 3e is significant
Example 3e: Is χ² significant?

To compare the chi-square value to the critical value I look up in the table the value for the chi-squared critical value when alpha = 0.05 and df = ____:

Therefore, since the absolute value of the test statistic is less than or equal to the critical value we (circle one):

reject the Ho fail to reject the Ho accept the Ho accept the Ha

And conclude….what?:

to review
To review:

When calculating a chi-squared value:

1. Find expected counts

2. Compare expected counts to observed counts

3. Calculate a χ² test statistic

4. Compare test statistic to critical value using table

5. Make a conclusion

If | χ²*| > χ²α, (r-1)(c-1) REJECT THE NULL: relationship exists

If | χ²*| ≤ χ²α, (r-1)(c-1) FAIL TO REJECT THE NULL: independent, no relationships exists

NEVER SAY ACCEPT THE NULL!!!!

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