16.4. Longhand calculation example#

In this worked example based on a small sample size, we can easily see how the key statistics for ANOVA are derived.

https://raw.githubusercontent.com/jillxoreilly/StatsCourseBook/main/images/regression4_CarCrash.jpg

Example:

A consumer protection group compares three types of front bumpers for a brand of automobile. A test is conducted by driving an automobile into a concrete wall at 15 miles per hour. The response variable is the amount of damage to the car, as measured by the repair costs, in hundreds of pounds. Due to the potentially large costs, the study conducts only three tests with each bumper type. The table below shows the raw data collected:

Bumper A

Bumper B

Bumper C

1

2

11

3

4

15

2

6

10

Mean

2

4

12

The grand mean across the three groups = 6

  • Specify the null and alternative hypotheses for an ANOVA test for differences in means.

  • How do we find the Within sum of squares?

  • Q: How do we find the Between sum of squares?

  • Find the mean square within and the mean square between by dividing by degrees of freedom. How do we find the degrees of freedom (df)?

  • Calculate the $F$-statistic and interpret.