Old versions of Minitab would not find the test statistic. This has been fixed in version 14.20, but go ahead and complete the tables by hand so that you understand the process better. Check yourself with Minitab.

- Label a column as
**obs**for observed frequencies - Enter the observed frequencies into that column
- Choose Stat / Tables / Chi-Square Goodness-of-Fit Test (one variable)
- The observed counts are in the
**obs**column - We are testing that the proportions are equal
- Click on Graphs and turn off all the graphs

- The observed counts are in the

A table is given. The table contains the expected frequency and a
contribution to chi-sq. The contribution to chi-sq is the
(obs-exp)^{2}/exp row of the table.

At the bottom, you will get a line that contains the df, test statistic, and p-value.

There are three options available in Minitab for dealing with probability distributions.

- Probability density - This option returns the probability density function for a specific value. The probability density function corresponds to the height of the curve at a specific value, so this would be useful for making a graph of the distribution.
- Cumulative probability - This option returns the area to the left of a critical value or test statistic. This would be useful for finding a p-value when you know the test statistic.
- Inverse cumulative probability - This option returns the critical value or test statistic that has the given area to the left. This would be useful for finding a critical value from the significance level.

Okay, so of those options, the cumulative probability is the one we need to use.

- Choose Calc / Probability Distributions / Chi-Square
- Select Cumulative Probability Distribution.
- Leave the non-centrality parameter at 0, we're not in graduate school.
- Enter the correct degrees of freedom. The degrees of freedom is one less than the number of categories.
- Click on Input Constant and then put your test statistic in as the constant.
- Click OK
- The area returned by Minitab is the area to the left, but this is a right tail test, so you need to manipulate the area to the left to find the area to the right.

Here are the assumptions. We have a chi-square distribution with 5 df and the test statistic was 8.4. When we used Minitab to find the area to the right of 8.4, it returned 0.864475. Since this is a right tail test, we want the area to the right of 8.4 and since the entire area under the curve is 1, we can subtract the area of on the left from 1 to get the area on the right. That is 1 - 0.864475 = 0.123555. That's our p-value.

By the way, the figure above was generated using Minitab. I created a column of the numbers between 0 and 12, counting by 0.1, using the Calc / Make patterened data / Simple set of numbers command. I then went into Calc / Probability distributions / Chi-square and chose the Probability density function. My input column was the column with the numbers between 0 and 12 and I stored the probability values into another column. I then did a scatterplot and cleaned up and annotated the results.

Also note that I stopped at 12 because it showed most of the interesting features of the graph before 12. However, the graph continues on to the right forever, approaching the x-axis (the probabilities get really close to zero).