Following are measurements of tensile strength in ksi (x) and Brinell hardness (
ID: 3434171 • Letter: F
Question
Following are measurements of tensile strength in ksi (x) and Brinell hardness (y) for 10 specimens of cold drawn copper. Assume that tensile strength and Brinell hardness follow a bivariate normal distribution.
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x y
106.2 35.0
106.3 37.2
105.3 39.8
106.1 35.8
105.4 41.3
106.3 40.7
104.7 38.7
105.4 40.2
105.5 38.1
105.1 41.6
a) Find a 95% confidence interval for p, the population correlation between tensile strength and Brinell hardness.
b) Can you conclude that p<0.3?
c) Can you conclude that p =/= 0?
Explanation / Answer
We assume that the data points are a random sample from a bivariate normal distribution. In computing the correlation coefficiient, we find
Sxx = Px ^2 i ? ( Pxi) ^2/n = 111579.79 ? 6 (1056.3)^2/10 = 2.821,
Sxy = Pxiyi ?( Pxi)(Pyi)/n = 41020.79?(388.4)(1056.3)/10 = ?5.902
so R < 0 and Syy = Py ^2 i ? ( Pyi) ^2/n = 15132.6 ? (388.4) ^2/1047.144 = 47.144. Hence
R 2 = S^ 2 xy / SxxSyy = (?5.902^)2 /2.821