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, find the projected test grade, to the nearest integer, for a student with a homework grade of 80.\nhomework grade (x) test grade (y)\n71 63\n72 75\n62 58\n80 77\n85 71\n74 65\n90 95\n83 68\n62 57

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, find the projected test grade, to the nearest integer, for a student with a homework grade of 80.\nhomework grade (x) test grade (y)\n71 63\n72 75\n62 58\n80 77\n85 71\n74 65\n90 95\n83 68\n62 57

Answer

Explanation:

Step1: Calculate means of x and y

Let (x = [71,72,62,80,85,74,90,83,62]), (y=[63,75,58,77,71,65,95,68,57]) (\bar{x}=\frac{71 + 72+62+80+85+74+90+83+62}{9}=\frac{689}{9}\approx76.6) (\bar{y}=\frac{63 + 75+58+77+71+65+95+68+57}{9}=\frac{639}{9}=71)

Step2: Calculate numerator and denominator for slope (m)

(numerator=\sum_{i = 1}^{9}(x_i-\bar{x})(y_i - \bar{y})) (=(71 - 76.6)(63 - 71)+(72-76.6)(75 - 71)+(62 - 76.6)(58 - 71)+(80 - 76.6)(77 - 71)+(85 - 76.6)(71 - 71)+(74 - 76.6)(65 - 71)+(90 - 76.6)(95 - 71)+(83 - 76.6)(68 - 71)+(62 - 76.6)(57 - 71)) (=(- 5.6)\times(-8)+(-4.6)\times4+(-14.6)\times(-13)+3.4\times6+8.4\times0+(-2.6)\times(-6)+13.4\times24+6.4\times(-3)+(-14.6)\times(-14)) (=44.8-18.4 + 189.8+20.4+0 + 15.6+321.6-19.2+204.4) (=769)

(denominator=\sum_{i = 1}^{9}(x_i-\bar{x})^2) (=(71 - 76.6)^2+(72-76.6)^2+(62 - 76.6)^2+(80 - 76.6)^2+(85 - 76.6)^2+(74 - 76.6)^2+(90 - 76.6)^2+(83 - 76.6)^2+(62 - 76.6)^2) (=(-5.6)^2+(-4.6)^2+(-14.6)^2+3.4^2+8.4^2+(-2.6)^2+13.4^2+6.4^2+(-14.6)^2) (=31.36 + 21.16+213.16+11.56+70.56+6.76+179.56+40.96+213.16) (=788.2)

(m=\frac{numerator}{denominator}=\frac{769}{788.2}\approx0.98\approx1.0)

Step3: Calculate y - intercept (b)

(b=\bar{y}-m\bar{x}=71-1.0\times76.6=- 5.6\approx - 5.6) The linear - regression equation is (y = 1.0x-5.6)

Step4: Predict test grade for (x = 80)

Substitute (x = 80) into (y=1.0x - 5.6) (y=1.0\times80-5.6=74.4\approx74)

Answer:

The linear regression equation is (y = 1.0x-5.6), and the projected test grade for a student with a homework grade of 80 is 74.