1. Chi-Squared Goodness of Fit
Chapter 14

2. Chi – Squared Distribution:
Is a family of distributions specified by the degree of freedom (df) that has the following properties:

3. Chi – Squared Basics
Chi – Square test measures how far you are from an expected value.
The formula:
with df =(n-1)
The smaller the X2 – value the better fit.
The larger the X2 – value the more likely you are to reject the null hypothesis.

4. Chi – Squared Test The null hypothesis for the X2 test is:
The alternate hypothesis is:
X2 must exceed a critical value in order to be statistically significant.

5. Finding the Critical Value
Use table C to determine.
If we had df = 5 and want a 5% significance level the critical value would be
In other words:
As we do the test we also will find a p-value for this test.

6. Conditions/ Assumptions for the Goodness of Fit test
1. SRS
2. Must have counted data
(not means or proportions)
3. All expected values must be greater than 5.

7. Back to the M&M’s
Is the distribution of your significantly different from the M&M® claim at a = 5%? How about the distribution of the entire class?

8. Example 2
ABC, NBC, and CBS are all interested in winning the am show TV market. The people at Quinnipiac Polling Center polled 62 people to see what show they watch in the am watch ABC, 32 watch CBS, and 11 watch NBC. Is this difference in viewership significant at the 10% level?

9. Example 3
A radio station claims that the music preferences of the listeners are distributed as shown. You randomly select 500 listenres and get the results shown in the chard on the right. Using a = 0.1, can you conclude that the radio’s station claim was correct?
Type Music
% Favoring
Classical 4
Country 36
Gospel 11
Oldies 2
Pop 18
Rock 29
Type Music
# favoring
Classical 8
Country
210
Gospel 72
Oldies 10
Pop 75
Rock
125

10. Goodness of Fit recap Test uses univariate data
Wants to see how well the observed counts “fit” what we expect the counts to be
Can use X2cdf function of the calculator to find p- values.
Based on df where df = number of categories – 1
Hypotheses is written in words (be sure to write in context)
Ho: the observed count equals the expected counts
Ha: the observed counts are not equal to the
expected counts