A mathematics teacher wanted to see the correlation between test scores and homework. The homework grade () and test grade ) 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 5055 59 46 76 74 87 90 87 80 90 89
To find the linear regression equation, we first need to calculate the mean of the homework grades () and test grades ():
Mean of :
= (55 + 59 + 76 + 87 + 80 + 90) / 6
= 447 / 6
= 74.5
Mean of :
= (46 + 46 + 74 + 90 + 87 + 89) / 6
= 432 / 6
= 72
Next, we need to calculate the sample covariance () and the variance of :
Sample Covariance () = Σ(( - ) * ( - )) / ( - 1)
= ((55 - 74.5)(46 - 72) + (59 - 74.5)(46 - 72) + (76 - 74.5)(74 - 72) + (87 - 74.5)(90 - 72) + (80 - 74.5)(87 - 72) + (90 - 74.5)(89 - 72)) / 5
= (342.75 + 214.5 - 6 - 313.5 + 27.75 + 296.25) / 5
= 561 / 5
= 112.2
Variance of :
= Σ( - )^2 / ( - 1)
= ((55 - 74.5)^2 + (59 - 74.5)^2 + (76 - 74.5)^2 + (87 - 74.5)^2 + (80 - 74.5)^2 + (90 - 74.5)^2) / 5
= (317.25 + 229.5 + 3.75 + 156.25 + 30.25 + 225) / 5
= 962 / 5
= 192.4
Now, we can find the slope of the linear regression equation:
= 112.2 / 192.4
≈ 0.6
Now we can find the y-intercept () of the linear regression equation:
= - ()
≈ 74.5 - (0.6 * 72)
≈ 74.5 - 43.2
≈ 31.3
Therefore, the linear regression equation is: ≈ 31.3 + 0.6
To estimate the homework grade () for a student with a test grade of 38, we plug in = 38 into the equation:
≈ 31.3 + 0.6(38)
≈ 31.3 + 22.8
≈ 54.1
Therefore, the estimated homework grade for a student with a test grade of 38 is 54.