The Chi-Square Distributions

 CHISQDIS.83p CHISQDIS.86p chisqdis.89p

The Chi-Square distributions can be used to analyze the variance of a normally distributed population. In particular, we can construct confidence intervals and perform hypothesis tests on such variances with this distribution. The distribution is also used to conduct various goodness of fit tests.

A Chi-Square distribution is based on a parameter known as the degrees of freedom n, where n is an integer greater than or equal to 1. Such a random variable is denoted by X ~ X2(n). The probability density function (pdf) is given by

f(x) = C x^(n / 2 - 1) e^[-x / 2] ,

for x >= 0, where C is a constant that depends on n.

For n >= 3, the graph of the X2(n) pdf is a non-symmetric, skewed, "Bell-Shaped" curve defined for x >= 0. These graphs attain a maximum value at x = n - 2. Morever, as n increases, the random variable (X2(n) - n) / Sqrt(2n) converges in distribution to the standard normal distribution Z ~ N(0,1).

Using the CHISQDIS Program

The CHISQDIS program can be used to compute probabilities such as P(X2(n) <=k), P(X2(n) >= k), or P(j <= X2(n) <= k). To execute the program, first enter 1, 2, or 3 to designate the type of probability you wish to compute. Next, enter the degrees of freedom followed by the lower and upper bounds j and k, with 0 <= j <= k, (or enter just the single bound k for a tail probability). The program next asks if you want to see a shaded graph of the pdf. If so, enter 1. If not, then enter 0. After the graph, we receive a display of the desired probability.

Example. Let X ~ X2(25). Find the probability that X lies between 20 and 30.

Solution. Call up the CHISQDIS program, enter 3 for MIDDLE PROB, then enter 25 for DEG. OF FREEDOM, 20 for LOWER BOUND, and 30 for UPPER BOUND. We see that P(20 <= X2(25) <= 30) = 0.5225363.

Other Features:

1. If you entered 1 to see a graph, then the partially shaded graph of the pdf initially appears, but then is replaced by the display of the computed probability. To see the graph again, press GRAPH. Then to see the probability value again, press CLEAR. On the TI-86, press CLEAR twice, or just press EXIT to remove the graph. To exit the graph on the TI-89, press HOME. Then press F5 to see the program output again.

2. If you initially enter either 1 or 2 to compute a tail probability, then both the left-tail and right-tail probabilities will be displayed in either case. However only the desired region will be shaded in the graph.

Exercises

1. Compute P(X2(20) <= 18) and P(X2(20) > 18).

2. Approximate P(X2(85) > 80) by converting to an approximate standard normal distribution.

Solutions

1. After calling up the CHSQDIS program, enter 1 for LEFT PROB, then enter 20 for DEG. OF FREEDOM, followed by 18 for BOUND. We see that P(X2(20) <= 18) = 0.41259 and P(X2(20) > 18) = 0.58741.

2. Because (X2(85) - 85)/Sqrt(2*85) is approximately Z ~ N(0,1), we have

P(X2(85) > 80) = P( (X2(85) - 85) / Sqrt(2*85) > (80 - 85) / Sqrt(2*85) ) ~ P(Z > -5 / Sqrt(170)).

Thus, in the NORMDIST program, enter 1 for RIGHT PROB, then enter 0 for MEAN, enter 1 for STANDARD DEV., and enter -5 / Sqrt(170) for BOUND. We see that P(X2(85) > 80) is approximately 0.6493189.

Note: With the CHSQDIST program, we can compute the value of P(X2(85) > 80) more precisely as 0.6330177.