Question

х Run a regression analysis on the following bivariate set of data with y as the response variable. y 28.2 50.9 26.9 60.8 19.2 46.9 23.1 28.6 28.7 69.3 30.4 58.1 40.6 77.3 25.6 60.3 23.1 22.5 23.1 46 19.2 37.2 34.2 74.1 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.) ri Based on the data, calculate the regression line (each value to three decimal places) y Predict what value (on average) for the response variable will be obtained from a value of 22.1 as the explanatory variable. Use a significance level of a = 0.05 to assess the strength of the linear correlation. What is the predicted response value? (Report answer accurate to one decimal place.) y-

Answer 1

ΣX = 322.3
ΣY = 632
ΣX * Y = 17900.06
ΣX2 = 9078.17
ΣY2 = 36591.4

((N + ΣΧΥ) - ΣΧΣΥ)/((ΝΣ(Χ2) - (ΣΧ)2] (ΝΣ(Y2) – (ΣΥ)2)

$r = ( ( 12 * 17900.06 ) - 322.3 * 632 ) / ( \sqrt{ ( 12 * 9078.17 ) - ( 322.3 )^{2} )( 12 * 36591.4 ) - ( 632 )^{2} })$
r = 0.784

Coefficient of Determination
= 0.614
Explained variation = 0.614* 100 = 61.4%

Equation of regression line is Ŷ = a + bX

b = (nΣΧΥ – ΣΧΣΥ)/(ηΣΧ? - (ΣΧ)2)


b = 2.195
a =( Σ Y - ( b * Σ X) ) / n
a =( 632 - ( 2.1948 * 322.3 ) ) / 12
a = -6.281
Equation of regression line becomes Ŷ = -6.281 + 2.195 X

When X = 22.1
Ŷ = -6.281 + 2.195 X
Ŷ = -6.281 + ( 2.195 * 22.1 )
Ŷ = 42.2