Question

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 hundredth. using this equation, find the projected test grade, to the nearest integer, for a student with a homework grade of 68.

Explanation:

Step1: Calculate the means of \(x\) and \(y\)

Let \(x=\{57,85,80,75,87,50,58,79\}\), \(y = \{62,75,76,68,89,38,45,64\}\)
\(\bar{x}=\frac{57 + 85+80+75+87+50+58+79}{8}=\frac{571}{8}=71.375\)
\(\bar{y}=\frac{62+75+76+68+89+38+45+64}{8}=\frac{517}{8}=64.625\)

Step2: Calculate \(S_{xx}\), \(S_{xy}\)

\(S_{xx}=\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}\)
\((57 - 71.375)^{2}+(85 - 71.375)^{2}+(80 - 71.375)^{2}+(75 - 71.375)^{2}+(87 - 71.375)^{2}+(50 - 71.375)^{2}+(58 - 71.375)^{2}+(79 - 71.375)^{2}\)
\(=(- 14.375)^{2}+13.625^{2}+8.625^{2}+3.625^{2}+15.625^{2}+(-21.375)^{2}+(-13.375)^{2}+7.625^{2}\)
\(=206.640625+185.640625 + 74.390625+13.140625+244.21875+456.890625+178.890625+58.140625=1417.953125\)

\(S_{xy}=\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})\)
\((57 - 71.375)(62 - 64.625)+(85 - 71.375)(75 - 64.625)+(80 - 71.375)(76 - 64.625)+(75 - 71.375)(68 - 64.625)+(87 - 71.375)(89 - 64.625)+(50 - 71.375)(38 - 64.625)+(58 - 71.375)(45 - 64.625)+(79 - 71.375)(64 - 64.625)\)
\(=(-14.375)(-2.625)+13.625\times10.375+8.625\times11.375+3.625\times3.375+15.625\times24.375+(-21.375)\times(-26.625)+(-13.375)\times(-19.625)+7.625\times(-0.625)\)
\(=37.734375+141.484375+98.109375+12.28125+381.8359375+569.109375+262.421875 - 4.765625=1508.2109375\)

Step3: Calculate the slope \(m\) and intercept \(b\)

The slope \(m=\frac{S_{xy}}{S_{xx}}=\frac{1508.2109375}{1417.953125}\approx1.06\)
The intercept \(b=\bar{y}-m\bar{x}=64.625-1.06\times71.375=64.625 - 75.6575=-11.03\)
The linear regression equation is \(y = 1.06x-11.03\)

Step4: Predict \(y\) when \(x = 68\)

Substitute \(x = 68\) into \(y=1.06x - 11.03\)
\(y=1.06\times68-11.03=72.08-11.03 = 61.05\approx61\)

Answer:

The linear regression equation is \(y = 1.06x-11.03\), and the projected test grade for \(x = 68\) is \(61\)