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, estimate the homework grade, to the nearest integer, for a student with a test grade of 44.
homework grade (x) test grade (y)
78 70
84 84
74 60
80 69
70 64
75 68
62 56

Explanation:

Step1: Calculate the means of x and y

Let $n = 7$.
$\bar{x}=\frac{78 + 84+74+80+70+75+62}{7}=\frac{523}{7}\approx74.71$
$\bar{y}=\frac{70 + 84+60+69+64+68+56}{7}=\frac{471}{7}\approx67.29$

Step2: Calculate the numerator and denominator for the slope (m)

The formula for the slope $m=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})}{\sum_{i=1}^{n}(x_i-\bar{x})^2}$

$\sum_{i = 1}^{7}(x_i-\bar{x})(y_i - \bar{y})=(78 - 74.71)(70 - 67.29)+(84 - 74.71)(84 - 67.29)+(74 - 74.71)(60 - 67.29)+(80 - 74.71)(69 - 67.29)+(70 - 74.71)(64 - 67.29)+(75 - 74.71)(68 - 67.29)+(62 - 74.71)(56 - 67.29)$
$=3.29\times2.71 + 9.29\times16.71+(- 0.71)\times(-7.29)+5.29\times1.71+(-4.71)\times(-3.29)+0.29\times0.71+(-12.71)\times(-11.29)$
$=8.8159+155.2359 + 5.1759+9.0459+15.4959+0.2059+143.4959$
$=337.4763$

$\sum_{i=1}^{7}(x_i-\bar{x})^2=(78 - 74.71)^2+(84 - 74.71)^2+(74 - 74.71)^2+(80 - 74.71)^2+(70 - 74.71)^2+(75 - 74.71)^2+(62 - 74.71)^2$
$=3.29^2+9.29^2+(-0.71)^2+5.29^2+(-4.71)^2+0.29^2+(-12.71)^2$
$=10.8241+86.3041+0.5041+27.9841+22.1841+0.0841+161.5441$
$=309.4327$

$m=\frac{337.4763}{309.4327}\approx1.09$

Step3: Calculate the y - intercept (b)

The formula for the y - intercept $b=\bar{y}-m\bar{x}$
$b = 67.29-1.09\times74.71$
$b=67.29 - 81.4339\approx - 14.14$

The linear regression equation is $y = 1.09x-14.14$

Step4: Estimate x when y = 44

We substitute $y = 44$ into the equation $y = 1.09x-14.14$
$44=1.09x-14.14$
$1.09x=44 + 14.14$
$1.09x=58.14$
$x=\frac{58.14}{1.09}\approx53$

Answer:

The linear regression equation is $y = 1.09x-14.14$. The estimated homework grade for a student with a test grade of 44 is 53.