# Confidence Interval Population Correlations

Exercise 1

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.

______________________________________________

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 , the population correlation between tensile strength and Brinell hardness.

b) Can you conclude that

c) Can you conclude that

Exercise 2

The article "Optimization of Medium Composition for Lipase Production by Candida rugosa NCIM 3462 Using Response Surface Methodology" describes a series of experiments in which lipase was produced from a bacterial culture. In each experiment, the rate of lipase production (in mol per ml enzyme per minute) and the cell mass (in g/L) were measured. The results are presented in the following table:

Cell mass (x) Lipase (y) Cell mass (x) Lipase (y) Cell mass (x) Lipase (y) Cell mass (x) Lipase (y)

4.5 2.06 3.98 2.1 3.65 2.2 4.15 3.75

4.68 2.1 4.72 2.75 4.23 2.3 4.3 3.15

5.4 3.15 3.41 2.8 4.1 2.4 4.9 5.1

5.45 4.1 4.8 4.6 5.03 4.75 5.23 5.04

4.2 2.2 3.6 2.5 4.19 3.15 5.4 4.96

4.12 3.2 4.95 4.1 4.4 3.9 4.85 5

4 2.85 3.25 2.15 3.92 3.2 5.1 4.92

4.41 4.5 4.4 4.4 3.5 2.1 4.94 4.98

Using SAS, create a scatter plot, find the least squares line (prediction line), Pearson's correlation coefficient (explain), complete a test of linearity based on your SAS output, and find/explain the coefficient of determination. Note: Print out of SAS code and output is needed.

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#### Solution Preview

Please see the attachment.

Exercise 1

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.

______________________________________________

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 , the population correlation between tensile strength and Brinell hardness.

libname mysas 'c:cdata';

data mysas.Brinell;

input x y;

datalines;

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

;

run;

proc corr data=mysas.Brinell outp=mysas.corr; *outp with pearson correlation coefficient;

var x y;

run;

data mysas.corr_ci;

set mysas.corr (rename=(x=corr) drop=y _name_);

retain n;

if _type_='N' then n=corr;

if _type_='CORR'and corr ^= 1;

fishersz=0.5*(log(1+corr)-log(1-corr)); *Fisher Z transformation;

sigmaz=1/sqrt(n-3); *variance;

l95=fishersz-1.96*sigmaz; *á=0.05, i.e. at 95% level;

u95=fishersz+1.96*sigmaz;

l95=(exp(2*l95)-1)/(exp(2*l95)+1); *inverse of Fisher Z ;

u95=(exp(2*u95)-1)/(exp(2*u95)+1); *transformation to get CI;

run;

proc print data=mysas.corr_ci;

run;

/*The CORR ...

#### Solution Summary

The confidence interval population correlations are examined/