question\na mathematics teacher wanted to see the correlation between test scores and homework. the homework…

question\na 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.\nhomework grade (x) test grade (y)\n55 46\n50 55\n59 46\n76 74\n87 90\n87 80\n90 89\nanswer attempt 2 out of 2\nregression equation:\nfinal answer:

question\na 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.\nhomework grade (x) test grade (y)\n55 46\n50 55\n59 46\n76 74\n87 90\n87 80\n90 89\nanswer attempt 2 out of 2\nregression equation:\nfinal answer:

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: $44$