Chi-square and F Distributions. Children of the Normal. Questions. What is the chi-square distribution? How is it related to the Normal? How is the chi-square distribution related to the sampling distribution of the variance?
Children of the Normal
z score squared
Make it Greek
What would its sampling distribution look like?
Minimum value is zero.
Maximum value is infinite.
Most values are between zero and 1;
most around zero.
What if we took 2 values of z2 at random and added them?
Same minimum and maximum as before, but now average should be a bit bigger.
Chi-square is the distribution of a sum of squares. Each squared deviation is taken from the unit normal: N(0,1). The shape of the chi-square distribution depends on the number of squared deviates that are added together.
The distribution of chi-square depends on 1 parameter, its degrees of freedom (df or v). As df gets large, curve is less skewed, more normal.
Sample estimate of population variance (unbiased).
Multiply variance estimate by N-1 to get sum of squares. Divide by population variance to stadnardize. Result is a random variable distributed as chi-square with (N-1) df.
We can use info about the sampling distribution of the variance estimate to find confidence intervals and conduct statistical tests.
Test the null that the population variance has some specific value. Pick alpha and rejection region. Then:
Plug hypothesized population variance and sample variance into equation along with sample size we used to estimate variance. Compare to chi-square distribution.
Test about variance of height of people in inches. Grab 30 people at random and measure height.
Note: 1 tailed test on small side. Set alpha=.01.
Mean is 29, so it’s on the small side. But for Q=.99, the value of chi-square is 14.257. Cannot reject null.
Note: 2 tailed with alpha=.01.
Now chi-square with v=29 and Q=.995 is 13.121 and also with Q=.005 the result is 52.336. N. S. either way.
We use to estimate . It can be shown that:
Suppose N=15 and is 10. Then df=14 and for Q=.025 the value is 26.12. For Q=.975 the value is 5.63.
In our applications, v2 will be larger than v1 and v2 will be larger than 2. In such a case, the mean of the F distribution (expected value) is
v2 /(v2 -2).
e.g. critical value of F at alpha=.05 with 3 & 12 df =3.49
Going to the F table with 15 and 15 df, we find that for alpha = .05 (1-tailed), the critical value is 2.40. Therefore the result is significant.