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HYPOTHESES - EXERCISE Testing an Hypothesis Regarding...

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EXERCISE: Testing an Hypothesis Regarding Pearson’s Data on Heights From the evidence of Figure 1 of Lecture 1, which shows two regression lines fitted to Pearson’s data, there seems to have been a significant increase in the heights of adult males from one generation to the next in late Victorian England. The object of this exercise is to test whether the increase is statistically significant. Let x denote the height of a father and y denote the height of his son. We may assume that these heights are normally distributed with means of E ( x ) = µ x and E ( y ) = µ y , with variances of V ( x ) = σ 2 x and V ( y ) = σ 2 y and with a covariance of C ( x, y ) = ρσ x σ x , which allows us to write x y N µ x µ y , σ 2 x ρσ x σ x ρσ x σ x σ 2 y . (1) We shall also assume that V ( x ) = V ( y ) = σ 2 , and we shall endeavour to test the hypothesis that E ( x ) = E ( y ) = µ , given a sample of N observations contained in the vectors x = [ x 1 , . . . , x N ] and y = [ y 1 , . . . , y N ] . On that basis, there is x y N µ x ι µ y ι , σ 2 I ρI ρI I , (2) where ι = [1 , 1 , . . . ,
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