Question

2. (13 pts.) The director of admissions of a small college selected 10 students at random from the new freshman class in a study to determine whether a student's GOA at the end of the freshman year can be predicted from the ACT test score. The data is GPA.csv. a) (4) Compute a linear regression to predict GPA based on the ACT test score. Y = 0.01 0147X + 2.908258 b) (2) Compute the residual standard deviation, s (2) Give a point estimate and a 95% confidence interval for the slope and intercept and interpret each of these in words. (Point estimate is another word for parameter estimate.) Why would someone be interested if 0 is included in the interval for β? c) d) (5) Perform a hypothesis test on whether ACT test score is associated with GPA.

Answer 1

Let us suppose that ACT test score is denoted with X and is treated as independent variable

and GPA is denoted with Y and is treated as dependent variable

a. From the given data sum of X= 269 sumof squares of X =7333

Sum Of Y = 31.812 Sum of squares of Y = 101.633

Sum of the product of X and Y = 856.726

Mean of X = X^-- = 26.9 Mean of Y = Y^-- = 3.1812

S.D of X = $\sigma$ _x= 3.1129 S.D of Y = $\sigma$ _y= 0.2080

covariance between X and y = 0.09832

Correlation coefficient between X and Y = r= 0.1518

Regression coefficient of Yon X = Beta coefficient =B= 0.010147

Regression equation of Yon X is given as (Y - mean of Y ) = B ( X- mean 0f X)

(Y- 3.1812) = 0.010147( X- 26.9)

Y= 0.010147X + 2.90825

b.compute the residual standard deviation

residual standard deviation = $\sqrt{}$ residual sum of square /n-1 = $\sqrt{}$ 0.4226/9 =0.2166

c. point and interval estimate of intercept and slope

point estimate of slope = B = 0.010147 and intercept = 2.90825

interval estimate of slope B = B+/- (t xs.e(B)) = 0.010147+/- (2.31x0.07553

=(-0.1643, 0.1846)

   Interval estimate of intercept = 2.90825+/- 0.5309 = (2.377, 3.43919)

d. test whether ACT test score is associated with GPA

set up H0 ; B = 0 i.e the regession coefficient is not significant

   H1 ; B not equal to 0 i.e the regression coefficient is significant it means ACT test score is influencing the GPT

  Under H0 the test statistic t= ( Estimated B - 0) /s.e (B) it FOllows t distribution with n-2 d.f

=0.010147/0.0738 = 0.1374

critical value of t for 8 degrees of freedom at 5 % level of significance = 2.31

we observe that t cal value = 0.1374 is less than t tab value = 2.31

hence we accept the null hypothesis i.e the regression coefficient is not significant at 5% level of significance

hence we conclude that ACT test score is not associated with GPA