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Chi-square and F Distributions

Children of the Normal

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Distributions

• There are many theoretical distributions, both continuous and discrete.

• We use 4 of these a lot: z (unit normal), t, chi-square, and F.

• Z and t are closely related to the sampling distribution of means; chi-square and F are closely related to the sampling distribution of variances.

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Chi-square Distribution (1)

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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.

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Chi-square (2)

What if we took 2 values of z2 at random and added them?

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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.

Same minimum and maximum as before, but now average should be a bit bigger.

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Chi-square 3The 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.

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Chi-square (4)

• The expected value of chi-square is df.– The mean of the chi-square distribution is its

degrees of freedom.

• The expected variance of the distribution is 2df. – If the variance is 2df, the standard deviation must

be sqrt(2df).

• There are tables of chi-square so you can find 5 or 1 percent of the distribution.

• Chi-square is additive. 2)(

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Distribution of Sample Variance

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Sample estimate of population variance (unbiased).

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Multiply variance estimate by N-1 to get sum of squares. Divide by population variance to normalize. 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.

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Testing Exact Hypotheses about a Variance

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variance has some specific value. Pick alpha and rejection region. Then:

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Plug hypothesized population variance and sample variance into equation along with sample size we used to estimate variance. Compare to chi-square distribution.

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Example of Exact Test

Test about variance of height of people in inches. Grab 30 people at random and measure height.

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HH Note: 1 tailed test on small side. Set alpha=.01.

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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.

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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.

Note: 2 tailed with alpha=.01.

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Confidence Intervals for the VarianceWe use to estimate . It can be shown that:2s

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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.

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Normality Assumption

• We assume normal distributions to figure sampling distributions and thus p levels.

• Violations of normality have minor implications for testing means, especially as N gets large.

• Violations of normality are more serious for testing variances. Look at your data before conducting this test. Can test for normality.

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The F Distribution (1)

• The F distribution is the ratio of two variance estimates:

• Also the ratio of two chi-squares, each divided by its degrees of freedom:

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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).

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F Distribution (2)

• F depends on two parameters: v1 and v2 (df1 and df2). The shape of F changes with these. Range is 0 to infinity. Shaped a bit like chi-square.

• F tables show critical values for df in the numerator and df in the denominator.

• F tables are 1-tailed; can figure 2-tailed if you need to (but you usually don’t).

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Testing Hypotheses about 2 Variances• Suppose

– Note 1-tailed.

• We find

• Then df1=df2 = 15, and

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sF 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.

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A Look Ahead

• The F distribution is used in many statistical tests– Test for equality of variances.– Tests for differences in means in ANOVA.– Tests for regression models (slopes

relating one continuous variable to another like SAT and GPA).

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Relations among Distributions – the Children of the Normal• Chi-square is drawn from the normal.

N(0,1) deviates squared and summed.• F is the ratio of two chi-squares, each

divided by its df. A chi-square divided by its df is a variance estimate, that is, a sum of squares divided by degrees of freedom.

• F = t2. If you square t, you get an F with 1 df in the numerator.

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