Chi-square and F Distributions

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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)
z score
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.
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Chi-square (2)
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.
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Chi-square 3
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.
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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.

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Distribution of Sample Variance
Sample estimate of population variance (unbiased).
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
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.
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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.
Note 1 tailed test on small side. Set alpha.01.
Mean is 29, so its 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 v29 and Q.995 is 13.121 and
also with Q.005 the result is 52.336. N. S.
either way.
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Confidence Intervals for the Variance
We use to estimate . It can be shown that
Suppose N15 and is 10. Then df14 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

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 dont).

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Testing Hypotheses about 2 Variances

• Suppose
• Note 1-tailed.
• We find
• Then df1df2 15, and

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.