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 35.
homework grade (x) test grade (y)
90 78
88 87
88 85
83 78
81 74
81 76
52 41
62 55
copy values for calculator
open statistics calculator
answer attempt 1 out of 3
regression equation:
final answer:

Explanation:

Step1: Calculate means of x and y

Let \(x = [90,88,88,83,81,81,52,62]\) and \(y=[78,87,85,78,74,76,41,55]\).
\(\bar{x}=\frac{90 + 88+88+83+81+81+52+62}{8}=\frac{525}{8}=65.625\)
\(\bar{y}=\frac{78 + 87+85+78+74+76+41+55}{8}=\frac{574}{8}=71.75\)

Step2: Calculate slope \(m\)

\[

$$\begin{align*} m&=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}\\ \sum_{i = 1}^{8}(x_{i}-\bar{x})(y_{i}-\bar{y})&=(90 - 65.625)(78-71.75)+(88 - 65.625)(87 - 71.75)+(88 - 65.625)(85 - 71.75)+(83 - 65.625)(78 - 71.75)+(81 - 65.625)(74 - 71.75)+(81 - 65.625)(76 - 71.75)+(52 - 65.625)(41 - 71.75)+(62 - 65.625)(55 - 71.75)\\ &=24.375\times6.25+22.375\times15.25+22.375\times13.25+17.375\times6.25+15.375\times2.25+15.375\times4.25+(- 13.625)\times(-30.75)+(-3.625)\times(-16.75)\\ &=152.34375+341.21875+296.46875+108.59375+34.59375+65.34375 + 419.90625+60.78125\\ &=1489.25\\ \sum_{i = 1}^{8}(x_{i}-\bar{x})^{2}&=(90 - 65.625)^{2}+(88 - 65.625)^{2}+(88 - 65.625)^{2}+(83 - 65.625)^{2}+(81 - 65.625)^{2}+(81 - 65.625)^{2}+(52 - 65.625)^{2}+(62 - 65.625)^{2}\\ &=24.375^{2}+22.375^{2}+22.375^{2}+17.375^{2}+15.375^{2}+15.375^{2}+(-13.625)^{2}+(-3.625)^{2}\\ &=594.140625+499.640625+499.640625+301.890625+236.390625+236.390625+185.640625+13.140625\\ &=2566.875\\ m&=\frac{1489.25}{2566.875}\approx0.6 \end{align*}$$

\]

Step3: Calculate y - intercept \(b\)

\(b=\bar{y}-m\bar{x}=71.75-0.6\times65.625=71.75 - 39.375 = 32.4\)
The regression equation is \(y = 0.6x+32.4\)

Step4: Predict test - grade for \(x = 35\)

Substitute \(x = 35\) into \(y=0.6x + 32.4\)
\(y=0.6\times35+32.4=21 + 32.4=53.4\approx53\)

Answer:

Regression Equation: \(y = 0.6x+32.4\)
Final Answer: \(53\)