Testing Goodness of Fit on the TI-83/84

See also:  A separate TI-89 procedure is also available.

TI-83/84 Program GOF

With the TI-83 you have a lot of work to do to compute χ² for a goodness-of-fit test. The TI-84 automates about half of that, but still you have to do some fussy computations. I’ve developed a program for the TI-83 or TI-84 that handles all the computations and graphs the curve for you.

Getting the Program

There are three methods to get the program into your calculator:

• If a classmate has the program on her calculator (any model TI-83/84), she can transfer it to yours with the short cable with headphone-style plugs that comes with all TI calculators. On your calculator, press [2nd x,T,θ,n makes LINK] [►] [ENTER], and then on hers press [2nd x,T,θ,n makes LINK] [3], select the program, then press [►] [ENTER].
• Or, download GOF.ZIP (20 KB, updated Jun 21, 2008), unzip it, and transfer file GOF.8XP to your calculator. (This requires TI-Connect or TI-GraphLink software and a cable.)
• Or, as a last resort, key in the program. See GOF.PDF and GOF_HINTS.HTM in the GOF.ZIP file.

Using the Program

Put the model in L1 and the observed numbers in L2. Then press [PROG], scroll to GOF, and press [ENTER] twice. The program will compute χ² and the p-value and show them on a graph; see examples below.

After the program runs, these lists are filled in for you automatically:

• L3 = E, expected frequencies
• L4 = O−E, deviations of observed from expected
• L5 = (O−E)²/E, contributions to χ²

Always check the expected counts in L3 to make sure that the requirements for the GOF test are met: no E can be less than 1, and no more than a fifth of the E’s can be less than 5.

Example 1: Fruit Flies

An example in Dabes & Janik's Statistics Manual (1999) had to do with the offspring of hybrid fruit flies; see figures at right. The null hypothesis H₀ is that the 9:3:3:1 model is good, and the alternative H₁ is that the model is bad. To compute the p-value, as always, you assume the model is good (assume H₀ is true) and then compute the probability of getting the sample you got, or a sample even further from the model. Use α = 0.05.

The test statistic χ² is a measure of how far the observations differ from the model. The p-value measures the chance of getting a random sample this different from H₀ if H₀ is actually true. You’ve already learned to compute them by hand; now you’ll do it the easy way.

Enter the model in L1.	[STAT] [1] brings up the stats edit screen.   Go to the L1 column heading and press [CLEAR] [ENTER]. Enter the numbers 9, 3, 3, and 1 in the first four rows of L1.
Now enter the observed values in L2.	Cursor to the L2 column heading and press [CLEAR] [ENTER]. Enter the numbers 120, 49, 36, 12 in the first four rows of L2.
Run the program.	Press [PRGM]. If GOF is in the first screen, press its number; otherwise scroll down to GOF and press [ENTER].   The home screen shows prgmGOF. Press [ENTER] to confirm that you want to run it.   The program reminds you that the model must be in L1 and the observed frequencies in L2. If you have entered data in those lists, press [9].

The program results are shown at right. The program sketches the χ² distribution and displays the important numbers:

• The left edge (low) is the χ² statistic for this test, 2.45. (up is the right-hand edge of the shaded area, always ∞ since χ² tests are always one-tailed to the right.)
• The degrees of freedom (df) is 1 less than the number of possible responses. Here, df = 4−1 = 3.
• The shaded area is the p-value. Here, pval = 0.4838.

The screen reminds you to check L3 to make sure that the requirements for a χ² test are met. None of the E’s can be less than 1, and no more than 20% of the E’s can be less than 5. Here, there are 4 categories, and 20% of 4 is 0.8, so not even one of the E’s can be less than 5.

Press [STAT] and if necessary scroll to display L3 through L5. As you see, all the E’s in L3 are ≥ 5, so the requirements are met and the GOF test is usable for this experiment.

Now draw your conclusions. The p-value is greater than α, so you fail to reject H₀. At the 0.05 significance level, you can’t say whether the 9:3:3:1 model is good or bad. (Some researchers will say “the model is not inconsistent with the data”.)

Do the expected numbers have any significance other than for checking requirements? Yes, they show how the sample would be distributed among the categories assuming H₀ is true.

Of course, a sample hardly ever matches the null hypothesis exactly. L4 is the deviations, the amounts by which the actual observed numbers are over or under the expected numbers in the null hypothesis. L5 is the contributions to χ² — for example, the deviation in category 1 is greater than the deviation in category 4, but it contributes much less to χ² because the expected value is much greater.

Example 2: Equal Class Registrations

Sometimes you want to know whether unequal frequencies are in fact significantly unequal. In that case your model is a series of 1’s, indicating equal ratios. Here’s an example adopted from Johnson & Kuby’s Just the Essentials of Elementary Statistics 3/e page 463.

Suppose 119 college students registered for seven sections of a course in these numbers: 18, 12, 25, 23, 8, 19, 14. At the 0.05 level, do the data indicate that the students had a preference for certain sections, or was each section equally likely to be chosen?

H₀ is that each section was equally likely to be chosen, and H₁ is that students had a preference. Your model for H₀ is equal ratios of 1:1:1:1:1:1:1 (one 1 for each of the seven categories). Enter this in L1, enter the observed numbers in L2, and run the program.

The results are shown below. As you can see, χ² = 12.9412 with 6 degrees of freedom, and the p-value is 0.04398. Check the expected numbers in L3 — they’re all ≥ 5, so the requirements are met.

Since the p-value is less than α, you conclude at the 0.05 significance level that students have unequal preferences among sections.

L5 shows that the fifth section contributed most to χ², with the third section contributing almost as much. Based on the signs in L4, you would suspect that section 3 is most popular and secton 5 is least popular. However, you can’t conclude that from just the χ² GOF test; other statistical procedures would be necessary.