1. Categorical-Quantitative Association
One-Way Anova
Categorical-Quantitative Association

2. Logic of ANOVA Hypothesis Testing
The explanatory variable identifies the groups we are interested in, and is the categorical variable.
The response variable is what we want to study relative to those groups, and is the quantitative variable.

3. Logic of ANOVA Hypothesis Testing
The null hypothesis states that there is no difference between the group means.
The alternative hypothesis states that there is a difference between the groups means.

4. Logic of ANOVA Hypothesis Testing
The hypotheses for the ANOVA test are:
: at least two of the population means are different
Another way of phrasing this is:
There is no difference between group means in the
population.
: There is a difference between at least two group means in the
population

5. Example: On-hold tolerance
Population: All callers put on hold by United Airlines
Question: Does the type of on-hold message affect how long a customer will stay on hold?
Notes:
We are interested one categorical variable and one quantitative: type of on-hold message (advertisements/muzak/classical music) and the length of time staying on hold.
Which is the explanatory variable? The response variable?
We consider the mean of each group: μa, μm, and μc.

6. Setting up a hypothesis test
H0 : μa = μm =μc. Ha : At least two of the population means are unequal. Significance Level: α = 0.05

7. The sample data
United Airlines randomly selected one out of every 1000 calls in a particular week. For each call, they randomly selected on of the three recorded messages to play and then measured the number of minutes that the caller remained on hold before hanging up (these calls were not answered on purpose). Below is a summary:

8. ANOVA ANOVA stands for analysis of variance.
Variance is the amount of variation of something.
The ANOVA hypothesis test is based on measurements of the variance within each group and the variance between the groups.

9. Considering different variances
The dot plots for each group

10. Considering different variances
Hypothetical example of different dot plots: what do you notice?

11. The F test statistic
For the ANOVA hypothesis test, the test statistic is called an F score. Large F yields low P-value.
Larger variance between groups results in a larger F score, which supports the alternative hypothesis.
Larger variance within groups results in a smaller F score, which supports the null hypothesis.

12. Finding the P-value using the F distribution
We always compute a right tail for the ANOVA hypothesis test.
On the TI-83/83 this would be Fcdf(a, 10^9, dfNumer, dfDenom).
As before, if the P-value is less than the significance, we reject the null hypothesis. Otherwise, we fail to reject.

13. ANOVA Table

14. Conclusion. Since P-value < 0.05, we reject the null hypothesis.
With a P-value of , we have evidence that a difference in the mean time customers are willing to remain on hold exists for the three types of on-hold messages.

15. ANOVA on TI 83/84
Given an F-score use the F-distribution to find area to the right.
Fcdf(a, 10^9, dfNumer, dfDenom)
dfNumer = # groups -1
dfDenom = total sample size - # groups
If using STAT > TESTS
Enter raw data for the groups into separate lists (L1, L2, etc)
STAT > TESTS > ANOVA and enter which lists are being used in this test
Output is F test statistic, P-value, and other information similar to the table output we saw earlier