Solve it with hands please Solve it with hands please Solve it with hands please

Question

Solve it with hands please Solve it with hands please   Solve it with hands please

Five randomly selected students took a math aptitude test before they began their statistics course. In the table below, the x_i column shows scores on the aptitude test. Similarly, the y_i column shows statistics grades. Assume the statistics grade y and math aptitude test x has a linear relationship of the form: y = b_0 + b_1x. Please do the following using the equations provided in the slides of linear regression. (1) Estimate the values of b_0 and b_1, as well as their standard errors. (2) Estimate the variance of residual error. (3) Estimate the R^2 of the fitted model. (4) Perform Significance Test for the slope b_1. (5) Comment on the model you fit.

Explanation / Answer

Back-up Theory

Let Xbar = average of aptitude maths scores and Ybar = average of statistics scores.

Sxx = sum(i = 1 to 5)(xi – Xbar); Syy = sum(i = 1 to 5)(yi – Ybar); Sxy = sum(i = 1 to 5)(xi – Xbar)(yi – Ybar).

Preparatory Calculations

Xbar = 78;   Ybar = 77; Sxx = 730;   Syy = 630; Sxy = 470

Part (1)

Estimate of b1 = Sxy/Sxx = 0.6438;     Estimate of b0 = Ybar – b1xXbar = 26.7806

Standard error of b1 = sq.rt{s2/Sxx} = 0.3866

Standard error of b0 = SE(b1)xsq.rt{(1/n)Sumxi2} = 30.518

where s2 = variance of residual error, as computed in Part (2).

Part (2)

Variance of residual error = s2 = Syy – b12.Sxx = 109.1324 ANSWER

Part (3)

Estimate of R2 = (Sxy)2/(SyyxSxx) = 0.4803 ANSWER

Part (4)

H0: b1 = 0 Vs H1: b1 0

Test Statistics = t = b1/SE(b1) = 1.6783

Under H0, t ~ tn -2 = t3

Upper 2.5% point of t3 is 3.182, being greater than 1.6783, H0 is accepted implying that b1 is not significant. ANSWER

Part (5)

Comments on the fit

An R2 of .0.4803 implies that only 48% of the variation in statistics score is explained by the aptitude maths score. Thus, a linear fit does not explain the relation between the two scores. This is further corroborated by a insignificant b1.

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