Computing Critical χ² on the TI-83/84/89

Summary:  Some statistical procedures require you to compute critical χ² or inverse chi-squared. You’re given the degrees of freedom and the significance level or area of the right-hand tail, and you have to determine the value of the χ² statistic that divides the area at the necessary point. This page explains how to do that, using your TI calculator.

See also:  Inferences about Standard Deviation of One Population gives some applications, as well as some non-calculator methods for finding critcal chi-squared.

Notation

χ²(df,rtail) is the critical value for the χ² distribution with df degrees of freedom and probability rtail. (In the context of a hypothesis test, rtail is α, the significance level of the test.)

In the illustration, rtail is the area of the right-hand tail, and the asterisk * marks the critical value χ²(df,rtail). The critical value of inverse χ² is the χ² value such that a higher value of χ² has only an rtail probability of occurring by chance.

You can compute critical χ² only for the right-hand tail, because the χ² distribution has no left-hand tail.

Caution:  Some textbooks write the function the other way, χ²(rtail,df). Since df is a whole number and rtail is a decimal between 0 and 1, you will be able to adapt.

TI-89 Procedure

The flash application, Stats/List Editor, can compute inverse χ² directly, for df and rtail area.

1. Press [F5] [2] [3] for inverse chi-squared.
2. In the “area” box, enter the area of the left-hand region, not the right-hand tail. Since the total area is 1, the area of the left-hand region is 1−rtail.
3. Enter the degrees of freedom, and press the [ENTER] key twice.

Example 1: What is the critical χ² for a 0.05 significance test with 13 degrees of freedom?

Solution: In the Stats/List Editor, press [F5] [2] [3] to bring up the dialog box. You need the area of the left-hand region, but 0.05 is the area of the right-hand tail. The area of the left-hand region is 1 minus that, so enter 1−.05 in the Area box. Enter 13 in the df box. Press [ENTER] twice. After a moment, the answer of 22.36 appears.

Answer: χ²(13,0.05) = 22.36

Try this with the examples below.

TI-83/84 Program INVCHI2

The TI-83 and TI-84 can’t compute inverse χ² natively, but I have written a program that adds this ability.

Getting the Program

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

• If you have the TI-Connect or TI-Graph Link software and a cable, download INVCHI2.ZIP (3 KB), unzip it, and transfer file INVCHI2.8xp to your calculator. (If nothing happens when you double-click the INVCHI2.ZIP file, you need a ZIP program; I recommend WinZip.)
• If a classmate already has the program on her calculator, you can transfer it to yours by using 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]. (It’s okay if you don’t have the same exact model, as long as each one is a TI-83 or TI-84.)
• If you don’t have the software and cable, and you can’t get the program from a classmate, you can key it in. For the complete source code of the program, download the INVCHI2.ZIP file as above, and then double-click on INVCHI2_SourceCode.htm within the ZIP file.

Using the Program

To use the program, press [PRGM], select INVCHI2, and press [ENTER] [ENTER]. When prompted, enter the number of degrees of freedom and the area of the right-hand tail. The program uses the calculator’s numerical equation solver, and you should expect a pause of a few seconds while the solution is computed. You can see a faint “working” indicator moving in the upper right corner of the screen.

Caution:  There’s no error checking of your inputs. If you enter an impossible number of degrees of freedom or an impossible area (≤0 or ≥1), the program will fail with an unhelpful message. Just hit [2nd MODE makes QUIT] and try again.

Example 1: What is the critical χ² for a 0.05 significance test with 13 degrees of freedom?

Solution: Run the INVCHI2 program. Enter 13 for df and .05 for right tail. In a few seconds you'll see the answer of 22.36.

Answer: χ²(13,0.05) = 22.36

More Examples

Example 2: What is χ²(40,0.01)? (In words, for 40 degrees of freedom, what is the critical value of χ² at the 0.01 significance level? Or, to say it another way, what value of χ² splits the curve between a 99% area on the left and a 1% area for the right-hand tail?)

Solution: On the TI-89, press [F5] [2] [3]; enter 1−.01 for area and 40 for df. Using the TI-83/84 program, enter 40 for df and 0.01 for area to right. Either way, the answer is 62.43.

Interpretation (1): if you draw a vertical line through the df=40 χ² curve at χ²=62.43, the area to the right of the line will be 1% or 0.01.

Interpretation (2): if you compute a χ² value greater than 62.43 for df=40, you will reject the null hypothesis at the 0.01 significance level.

Example 3: What is χ²(3,0.01)?

iAnswer: 11.34487