Question

run a regression analysis on the following data set with y as the response variable.

x	y
62	77.3
56.9	73.1
89.3	81.7
86.4	79.8
76	77.5
79.3	80.2
81.7	79.2
67.2	78.2

what is the predicted value of the response variable when the explanatory variable has value 69.6? y =
round to 1 decimal place.
check answer

Explanation:

Step1: Calculate sums

Let \(n = 8\). Calculate \(\sum x\), \(\sum y\), \(\sum x^2\), \(\sum xy\).
\(\sum x=62 + 56.9+89.3+86.4+76+79.3+81.7+67.2 = 608.8\)
\(\sum y=77.3 + 73.1+81.7+79.8+77.5+80.2+79.2+78.2 = 636\)
\(\sum x^{2}=62^{2}+56.9^{2}+89.3^{2}+86.4^{2}+76^{2}+79.3^{2}+81.7^{2}+67.2^{2}\)
\(=3844+3237.61+7974.49+7464.96+5776+6288.49+6674.89+4515.84 = 45777.37\)
\(\sum xy=62\times77.3+56.9\times73.1+89.3\times81.7+86.4\times79.8+76\times77.5+79.3\times80.2+81.7\times79.2+67.2\times78.2\)
\(=4792.6+4159.39+7395.81+6894.72+5890+6359.86+6470.64+5255.04 = 45217.06\)

Step2: Calculate slope \(b_1\)

The formula for the slope \(b_1\) of the regression - line is \(b_1=\frac{n\sum xy-\sum x\sum y}{n\sum x^{2}-(\sum x)^{2}}\)
\(n\sum xy=8\times45217.06 = 361736.48\)
\(\sum x\sum y=608.8\times636 = 387296.8\)
\(n\sum x^{2}=8\times45777.37 = 366218.96\)
\((\sum x)^{2}=608.8^{2}=370637.44\)
\(b_1=\frac{361736.48 - 387296.8}{366218.96-370637.44}=\frac{-25560.32}{-4418.48}\approx5.785\)

Step3: Calculate intercept \(b_0\)

The formula for the intercept \(b_0\) is \(b_0=\bar{y}-b_1\bar{x}\), where \(\bar{x}=\frac{\sum x}{n}=\frac{608.8}{8}=76.1\) and \(\bar{y}=\frac{\sum y}{n}=\frac{636}{8}=79.5\)
\(b_0 = 79.5-5.785\times76.1\)
\(=79.5 - 440.2385=-360.7385\)

The regression equation is \(y = b_0 + b_1x=-360.7385+5.785x\)

Step4: Predict value

When \(x = 69.6\), \(y=-360.7385+5.785\times69.6\)
\(y=-360.7385 + 403.636=42.9\)

Answer:

\(77.9\)