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2. Chi-square and F DistributionsChildren of the 3. Distributions There are many theoreticaldistributions, both continuous anddiscrete. We use 4 of these a lot: z (unit normal),t, chi-square, and F. Z and t are closely related to thesampling distribution of means; chi-squareand F are closely related to thesampling distribution of 4. Chi-square Distribution (1)z X X= ( - ) ; z = (X -m ) ; z = ( y - m)ssSDz = y -m222 ( )sz scorez score squaredz2 = c Make it Greek2(1)What would its sampling distribution look like?Minimum value is zero.Maximum value is infinite.Most values are between zero and 1;most around 5. Chi-square (2)What if we took 2 values of z2 at random and added them?z = ( y -m ) ; z = ( y - )2222 222 22 11sms2= ( y - ) + ( y - ) = z 2+ z22 12c m222 1(2)smsSame minimum and maximum as before, but now averageshould 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 distributiondepends on the number of squared deviates that areadded 6. Chi-square 3The distribution of chi-square dependson 1 parameter, its degrees of freedom(df or v). As df gets large, curve is lessskewed, more 7. Chi-square (4) The expected value of chi-square is df. The mean of the chi-square distribution is itsdegrees of freedom. The expected variance of the distribution is2df. If the variance is 2df, the standard deviation mustbe sqrt(2df). There are tables of chi-square so you can find5 or 1 percent of the distribution. Chi-square is additive. 22(v1 v2 ) v1 v2 c = c + c +( )2( ) 8. Distribution of SampleVariance( y -y)212-= NsSample estimate of population variance(unbiased).c N s222( 1)( 1)sN= - -Multiply variance estimate by N-1 toget sum of squares. Divide bypopulation variance to normalize.Result is a random variable distributedas chi-square with (N-1) df.We can use info about the sampling distribution of thevariance estimate to find confidence intervals andconduct statistical 9. Testing Exact Hypothesesabout a Variance202H0 :s =s Test the null that the populationvariance has some specific value. Pickalpha and rejection region. Then:c N s2022( 1)( 1)sN= - -Plug hypothesized populationvariance and sample variance intoequation along with sample size weused to estimate variance. Compareto chi-square 10. Example of Exact TestTest about variance of height of people in inches. Grab 30people at random and measure height.H : s 6.25; H : s s16; 5.8; 16; 2 1.72 221 1 N = s = N = s =F s 5.83.41Going to the F table with 15= 1 = =s1.7222and 15 df, we find that for alpha= .05 (1-tailed), the criticalvalue is 2.40. Therefore theresult is 16. A Look Ahead The F distribution is used in manystatistical tests Test for equality of variances. Tests for differences in means in ANOVA. Tests for regression models (slopesrelating one continuous variable to anotherlike SAT and GPA) 17. 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, eachdivided by its df. A chi-square dividedby its df is a variance estimate, that is,a sum of squares divided by degrees offreedom. F = t2. If you square t, you get an Fwith 1 df in the numerator.2(v) v t = F(1, )


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