A sociologist is interested in studying the IQs of teachers from low income area
ID: 3331700 • Letter: A
Question
A sociologist is interested in studying the IQs of teachers from low income areas of a major city. Six schools were randomly chosen from low income areas and from each of these schools, five teachers were randomly chosen. The following table summarizes the mean IQ for each of these schools (NOTE: These numbers are all made up and are not intended to reflect teachers' true IQ scores). School 1 2 3 45 6 Mean 97 99 94 109 98 103 (a) If MSE-40, is there significant variability in average IQ among schools in low income areas (use a=0.01)? (b) Estimate all variance components. (c) How much power does this study have if the true variances were such that 2..2 and n were increased to 10? (d) Suppose the national average IQ for teachers is 105. Test the null hypothesis that the average IQ of these teachers is not lower than the national average ( a 0.05 )Explanation / Answer
a.The grand sample mean is 100. This means the treatment sum of squares is
SSTtrt= 5(32+ 12+ 62+ 92+ 22+ 32) = 700
The F test is equal to (700/5)/40 = 3.5. The critical F value for 5 and 24 degrees of freedom is 3.90. Since this is larger than the test statistic we do not reject. There is not sucient evidence to reject the null hypothesis that the school variance is zero
b.The variance estimate for the error variance (between teachers) is 40 and the school variance is (140-40)/5 = 20
c.If 22=2(as is observed here) and n were 10, the value of = 1 + 10(.5) = 2.45 and the degrees of freedom error are 54. For=.01, this equates to= 25% or power of 75%. This means that this experiment withn= 5 is very underpowered to detect such a small dierence as that observed in this experiment.
d.In class, we showed that in a random eects model, the MSTtrt should be used in place of theMSEwhen computing the standard error of the grand mean. The test statistic is100-105p140/30=-2.31We have5 (not N-a=24)degrees of freedom so the critical t value is -2.015. Since this is smaller, we reject the null hypothesis. It does appear that the average IQ in these schools is lower than the national average.5
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