1. Test for Goodness of Fit
Section 14.1

2. Chi-Square test for goodness of fit
How can you test to see if your bag of M&M’s differs from the stated proportions of colors?
You could use six separate one proportion z tests but that wouldn’t compare how your proportions overall matched with the stated proportions.

3. Example 14.1
Chi-square statistic
Degrees of freedom

4. The Chi-Square Distributions
The chi-square distributions are a family of distributions that take only positive values and are skewed to the right.
A specific chi-square distribution is specified by one parameter, called the degrees of freedom.

5. Properties The total area under a chi square curve is equal to 1.
Each chi-square curve begins at 0 on the horizontal axis, increases to a peak, and then approaches the horizontal axis asymptotically from above.
Each chi-square curve is skewed to the right. As the number of degrees of freedom increase, the curve becomes more and more symmetrical and looks more like a normal curve.

6. Goodness of Fit Test
A goodness of fit test is used to help determine whether a population has a certain hypothesized distribution, expressed as percents of population members falling into various outcome categories.
The null hypothesis is that the actual population percents are equal to the hypothesized percentages.

7. Chi-Square Test Statistic
The chi-square test statistic is:
The degrees of freedom is n-1

8. When can you use this test?
You may use this test with critical values from the chi-square distribution when all individual expected counts are at least 1 and no more than 20% of the expected counts are less than 5.

9. Section 14.1 Exercises
Page 846, 14.1 – 14.10

10. Conducting inference by simulation
Step 1: Establish a correspondence between random numbers and categories.
Step 2: Determine a sample size n.
Step 3: Randomly generate n numbers and count the numbers that fall into each category.
Step 4: Perform a chi-square goodness of fit test.

11. Follow-up analysis
Compare the observed counts with the expected counts.
Look for the largest component of chi-square.