a mathematics teacher wanted to see the correlation between test scores and homework. the homework grade (x)…

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 64.\nhomework grade (x) test grade (y)\n59 45\n82 81\n74 59\n61 58\n83 78\n83 75\n85 73\n71 56\n88 80\nanswer\nregression equation: \nfinal answer:

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 64.\nhomework grade (x) test grade (y)\n59 45\n82 81\n74 59\n61 58\n83 78\n83 75\n85 73\n71 56\n88 80\nanswer\nregression equation: \nfinal answer:

Answer

Explanation:

Step1: Calculate the means

Let $x$ be the homework - grade and $y$ be the test - grade. $n = 9$ $\bar{x}=\frac{59 + 82+74 + 61+83+83+85+71+88}{9}=\frac{686}{9}\approx76.2$ $\bar{y}=\frac{45 + 81+59+58+78+75+73+56+80}{9}=\frac{615}{9}\approx68.3$

Step2: Calculate the numerator and denominator for the slope $m$

The formula for the slope $m$ of the regression line is $m=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}$ $\sum_{i = 1}^{9}(x_{i}-\bar{x})(y_{i}-\bar{y})=(59 - 76.2)(45 - 68.3)+(82 - 76.2)(81 - 68.3)+(74 - 76.2)(59 - 68.3)+(61 - 76.2)(58 - 68.3)+(83 - 76.2)(78 - 68.3)+(83 - 76.2)(75 - 68.3)+(85 - 76.2)(73 - 68.3)+(71 - 76.2)(56 - 68.3)+(88 - 76.2)(80 - 68.3)$ $=(- 17.2)(-23.3)+(5.8)(12.7)+(-2.2)(-9.3)+(-15.2)(-10.3)+(6.8)(9.7)+(6.8)(6.7)+(8.8)(4.7)+(-5.2)(-12.3)+(11.8)(11.7)$ $=400.76+73.66 + 20.46+156.56+65.96+45.56+41.36+63.96+138.06$ $=1006.3$ $\sum_{i = 1}^{9}(x_{i}-\bar{x})^{2}=(59 - 76.2)^{2}+(82 - 76.2)^{2}+(74 - 76.2)^{2}+(61 - 76.2)^{2}+(83 - 76.2)^{2}+(83 - 76.2)^{2}+(85 - 76.2)^{2}+(71 - 76.2)^{2}+(88 - 76.2)^{2}$ $=(-17.2)^{2}+(5.8)^{2}+(-2.2)^{2}+(-15.2)^{2}+(6.8)^{2}+(6.8)^{2}+(8.8)^{2}+(-5.2)^{2}+(11.8)^{2}$ $=295.84+33.64+4.84+231.04+46.24+46.24+77.44+27.04+139.24$ $=901.6$ $m=\frac{1006.3}{901.6}\approx1.1$

Step3: Calculate the y - intercept $b$

The formula for the y - intercept $b$ is $b=\bar{y}-m\bar{x}$ $b = 68.3-1.1\times76.2$ $b = 68.3 - 83.82=-15.52\approx - 15.5$ The linear regression equation is $y = 1.1x-15.5$

Step4: Solve for $x$ when $y = 64$

$64=1.1x-15.5$ $1.1x=64 + 15.5$ $1.1x=79.5$ $x=\frac{79.5}{1.1}\approx72$

Answer:

Regression Equation: $y = 1.1x-15.5$ Final Answer: $72$