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# Chris Morgan, MATH G160 [email protected] April 13, 2012 Lecture 30 - PowerPoint PPT Presentation

Chris Morgan, MATH G160 [email protected] 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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April 13, 2012

Lecture 30

Chapter 2.4:

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

• 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

• 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?

• 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?

• 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?

• 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

• 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

• 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)

Chi-Square (χ²) Distribution Critical Values

The first row is the alpha level

The first column is the number of df

• 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) )

Recall the equation for expected counts:

Recall the equation for chi-square:

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 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?:

• 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) )

Recall the equation for expected counts:

Recall the equation for chi-square:

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 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?:

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!!!!