Question

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, estimate the homework grade, to the nearest integer, for a student with a test grade of 38.
homework grade (x) test grade (y)
55 46
50 55
59 46
76 74
87 90
87 80
90 89
answer attempt 2 out of 2
regression equation:
final answer:

Explanation:

Step1: Calculate means of x and y

Let $x$ be homework - grade and $y$ be test - grade.
$\bar{x}=\frac{55 + 50+59+76+87+87+90}{7}=\frac{504}{7}=72$
$\bar{y}=\frac{46 + 55+46+74+90+80+89}{7}=\frac{480}{7}\approx68.6$

Step2: Calculate the slope (b)

$b=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}$
$(x_1-\bar{x})(y_1 - \bar{y})=(55 - 72)(46-68.6)=(-17)(-22.6) = 384.2$
$(x_2-\bar{x})(y_2 - \bar{y})=(50 - 72)(55 - 68.6)=(-22)(-13.6)=299.2$
$(x_3-\bar{x})(y_3 - \bar{y})=(59 - 72)(46 - 68.6)=(-13)(-22.6)=293.8$
$(x_4-\bar{x})(y_4 - \bar{y})=(76 - 72)(74 - 68.6)=4\times5.4 = 21.6$
$(x_5-\bar{x})(y_5 - \bar{y})=(87 - 72)(90 - 68.6)=15\times21.4 = 321$
$(x_6-\bar{x})(y_6 - \bar{y})=(87 - 72)(80 - 68.6)=15\times11.4 = 171$
$(x_7-\bar{x})(y_7 - \bar{y})=(90 - 72)(89 - 68.6)=18\times20.4 = 367.2$
$\sum_{i = 1}^{7}(x_{i}-\bar{x})(y_{i}-\bar{y})=384.2+299.2+293.8+21.6+321+171+367.2 = 1858$
$(x_1-\bar{x})^2=(55 - 72)^2=(-17)^2 = 289$
$(x_2-\bar{x})^2=(50 - 72)^2=(-22)^2 = 484$
$(x_3-\bar{x})^2=(59 - 72)^2=(-13)^2 = 169$
$(x_4-\bar{x})^2=(76 - 72)^2=4^2 = 16$
$(x_5-\bar{x})^2=(87 - 72)^2=15^2 = 225$
$(x_6-\bar{x})^2=(87 - 72)^2=15^2 = 225$
$(x_7-\bar{x})^2=(90 - 72)^2=18^2 = 324$
$\sum_{i = 1}^{7}(x_{i}-\bar{x})^{2}=289+484+169+16+225+225+324 = 1732$
$b=\frac{1858}{1732}\approx1.1$

Step3: Calculate the y - intercept (a)

$a=\bar{y}-b\bar{x}$
$a = 68.6-1.1\times72$
$a=68.6 - 79.2=-10.6$
The regression equation is $y = 1.1x-10.6$

Step4: Estimate x when y = 38

$38=1.1x-10.6$
$1.1x=38 + 10.6$
$1.1x=48.6$
$x=\frac{48.6}{1.1}\approx44$

Answer:

Regression Equation: $y = 1.1x-10.6$
Final Answer: $44$