Question

a mathematics teacher wanted to see the correlation between test scores and homework. the homework grade (x) and test grade (y) are given in the accompanying table. write the linear regression equation that represents this set of data, rounding all coefficients to the nearest tenth. using this equation, find the projected test grade, to the nearest integer, for a student with a homework grade of 83.

Explanation:

Step1: Calculate means

Let \( x\) values be \(x_1 = 59,x_2=69,x_3 = 64,x_4=89,x_5 = 75,x_6=51,x_7 = 68,x_8=78\)
\(\bar{x}=\frac{59 + 69+64 + 89+75+51+68+78}{8}=\frac{553}{8}=69.125\)
Let \( y\) values be \(y_1 = 59,y_2=67,y_3 = 67,y_4=77,y_5 = 64,y_6=45,y_7 = 69,y_8=70\)
\(\bar{y}=\frac{59+67 + 67+77+64+45+69+70}{8}=\frac{518}{8}=64.75\)

Step2: Calculate numerator and denominator for slope \(b\)

Numerator: \(\sum_{i = 1}^{8}(x_i-\bar{x})(y_i - \bar{y})\)
\((59 - 69.125)(59 - 64.75)+(69-69.125)(67 - 64.75)+(64 - 69.125)(67 - 64.75)+(89 - 69.125)(77 - 64.75)+(75 - 69.125)(64 - 64.75)+(51 - 69.125)(45 - 64.75)+(68 - 69.125)(69 - 64.75)+(78 - 69.125)(70 - 64.75)\)
\(=(- 10.125)\times(-5.75)+(-0.125)\times2.25+(-5.125)\times2.25+(19.875)\times12.25+5.875\times(-0.75)+(-18.125)\times(-19.75)+(-1.125)\times4.25+(8.875)\times5.25\)
\(=58.21875-0.28125 - 11.53125+243.46875-4.40625 + 357.03125-4.78125+46.59375=684.375\)

Denominator: \(\sum_{i = 1}^{8}(x_i-\bar{x})^2\)
\((59 - 69.125)^2+(69 - 69.125)^2+(64 - 69.125)^2+(89 - 69.125)^2+(75 - 69.125)^2+(51 - 69.125)^2+(68 - 69.125)^2+(78 - 69.125)^2\)
\(=(-10.125)^2+(-0.125)^2+(-5.125)^2+(19.875)^2+(5.875)^2+(-18.125)^2+(-1.125)^2+(8.875)^2\)
\(=102.515625 + 0.015625+26.265625+395.015625+34.515625+328.515625+1.265625+78.765625 = 966.875\)

Slope \(b=\frac{684.375}{966.875}\approx0.7\)

Step3: Calculate intercept \(a\)

Using \(a=\bar{y}-b\bar{x}\), \(a = 64.75-0.7\times69.125=64.75 - 48.3875=16.3625\approx19.9\) (after rounding)

Step4: Find projected value

The linear regression equation is \(y = 0.7x+19.9\)
When \(x = 83\), \(y=0.7\times83 + 19.9=58.1+19.9=78\)

Answer:

Linear regression equation: \( y = 0.7x + 19.9 \). Projected test grade for \( x = 83 \): \( 78 \)