10.2 Chi-Square:Goodness of Fit

Chi-Square: Goodness of Fit• We are continuing to work with the Chi-Square

distribution, however we are shifting gears• The goodness of fit test is used to check

whether the observed data counts confirm the expected distribution of counts into different categories.• We are asking whether a population follows a specified

distribution.

Hypotheses• The population fits the given distribution• The population has a different distribution

Expected Frequency• The expected frequency is the number of data

from the sample, in theory, that would fall into each category.

Sample Test Statistic

• Degree of Freedom: • number of categories in the distribution

• Larger values of the sample statistic indicate greater differences between the proposed distribution and the distribution followed by the sample. The larger the statistic, the stronger the evidence to reject the null hypothesis, thus goodness-of-fit tests are always right-tailed tests.

Sample Test Statistic

To find in the calculator• Enter the observed counts in L1• Enter the expected counts in L2• Define L3 to be (L1 – L2)2/L2• Press ENTER, 2nd, MODE• Hit 2nd, STAT, tab over to MATH and choose 5:sum• Hit 2nd 3 Enter

P-Value

To find P-Value in the calculator• Hit 2nd, VARS• Tab down to cdf, press ENTER• cdf ( test stat, E99, d.f.)• ENTER

How to Test for Goodness of Fit

1. State the hypotheses and the level of significance

2. Compute the expected frequency for each category

3. Compute the sample test statistic 4. Find the P-Value5. Conclude the test6. Interpret the results