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 hundredth. using this equation, find the projected test grade, to the nearest integer, for a student with a homework grade of 68.
homework grade (x) test grade (y)
57 62
85 75
80 76
75 68
87 89
50 38
58 45
79 64
copy values for calculator
open statistics calculator
answer
regression equation:
final answer:

Explanation:

Step1: Calculate sums

Let \(n = 9\) (number of data - points).
Calculate \(\sum x\), \(\sum y\), \(\sum x^{2}\), \(\sum xy\):
\(\sum x=57 + 85+80 + 75+87+50+58+79 = 671\)
\(\sum y=62 + 75+76+68+89+38+45+64 = 517\)
\(\sum x^{2}=57^{2}+85^{2}+80^{2}+75^{2}+87^{2}+50^{2}+58^{2}+79^{2}=57^{2}+7225 + 6400+5625+7569+2500+3364+6241=44298\)
\(\sum xy=(57\times62)+(85\times75)+(80\times76)+(75\times68)+(87\times89)+(50\times38)+(58\times45)+(79\times64)=3534+6375+6080+5100+7743+1900+2610+5056 = 42498\)

Step2: Calculate slope \(m\)

The formula for the slope \(m\) of the regression line is \(m=\frac{n\sum xy-\sum x\sum y}{n\sum x^{2}-(\sum x)^{2}}\)
\[

$$\begin{align*} m&=\frac{9\times42498 - 671\times517}{9\times44298-(671)^{2}}\\ &=\frac{382482-346907}{398682 - 450241}\\ &=\frac{35575}{- 51559}\approx1.07 \end{align*}$$

\]

Step3: Calculate y - intercept \(b\)

The formula for the y - intercept \(b\) is \(b=\overline{y}-m\overline{x}\), where \(\overline{x}=\frac{\sum x}{n}=\frac{671}{9}\approx74.56\) and \(\overline{y}=\frac{\sum y}{n}=\frac{517}{9}\approx57.44\)
\[

$$\begin{align*} b&=57.44-1.07\times74.56\\ &=57.44 - 80.7892\\ &\approx - 23.35 \end{align*}$$

\]

The regression equation is \(y = 1.07x-23.35\)

Step4: Predict test grade

When \(x = 68\), \(y=1.07\times68-23.35=72.76-23.35 = 49.41\approx49\)

Answer:

Regression Equation: \(y = 1.07x - 23.35\)
Final Answer: 49