Question

Run a regression analysis on the following bivariate set of data with y as the response variable.

x	y
73	16.1
80	14.9
72.5	8.5
55.8	33.6
54.6	23.4
76.6	26.2
74.6	19.1
40.2	40.6
58.7	25.8

Find the correlation coefficient and report it accurate to three decimal places.


What proportion of the variation in y can be explained by the variation in the values of x? Report answer as a percentage accurate to one decimal place. (If the answer is 0.84471, then it would be 84.5%...you would enter 84.5 without the percent symbol.)
%

Based on the data, calculate the regression line (each value to three decimal places)

ˆyy^ =  x +

Predict what value (on average) for the response variable will be obtained from a value of 75.3 as the explanatory variable. Use a significance level of α=0.05α=0.05 to assess the strength of the linear correlation.

What is the predicted response value? (Report answer accurate to one decimal place.)
ˆyy^ (y hat) =

Answer 1

X Values
∑ = 586
Mean = 65.111
∑(X - M_x)² = SS_x = 1419.389

Y Values
∑ = 208.2
Mean = 23.133
∑(Y - M_y)² = SS_y = 778.88

X and Y Combined
N = 9
∑(X - M_x)(Y - M_y) = -841.703

R Calculation
r = ∑((X - M_y)(Y - M_x)) / √((SS_x)(SS_y))

r = -841.703 / √((1419.389)(778.88)) = -0.801

As r=-0.801, so r^2=0.642

Which means 64.2% of variation in y is explained by x

Sum of X = 586
Sum of Y = 208.2
Mean X = 65.1111
Mean Y = 23.1333
Sum of squares (SS_X) = 1419.3889
Sum of products (SP) = -841.7033

Regression Equation = ŷ = bX + a

b = SP/SS_X = -841.7/1419.39 = -0.593

a = M_Y - bM_X = 23.13 - (-0.59*65.11) = 61.744

ŷ = -0.593X + 61.744

For x=75.3,

ŷ = (-0.593*75.3) + 61.744=17.1