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

Copyright © 2007–2008 by Stan Brown, Oak Road Systems

Copyright © 2007–2008 by Stan Brown, Oak Road Systems

**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:**
You may not need to compute the critical value of χ².
Inferences about One-Pop. Standard Deviation on the TI-83/84 gives a program for
estimating stndard deviation σ or variance σ²,
as well as doing hypothesis tests on σ.

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.

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

- Press [
`F5`

] [`2`

] [`3`

] for inverse chi-squared. - 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*. - 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.

The TI-83 and TI-84 can’t compute inverse χ² natively, but I have written a program uses the TI-83/84 equation solver to add this ability.

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 INVCHI2.ZIP (17 KB, updated Jun 22, 2008), unzip it, and transfer file INVCHI2.8XP to your calculator. (This requires TI-Connect or TI-Graph Link software software and a cable.)
- Or, as a last resort, key in the program. See INVCHI2.PDF and INVCHI2_HINTS.HTM in the INVCHI2.ZIP file.

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

**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)?

**Answer**: 11.34487

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This page is used in instruction at Tompkins Cortland Community College in Dryden, New York; it’s not an official statement of the College. Please visit www.tc3.edu/instruct/sbrown/ to report errors or ask to copy it.

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