The following table contains ACT scores and GPAs (grade point averages) for eigh

Question

The following table contains ACT scores and GPAs (grade point averages) for eight college students. Grade point averages are based on a four-point scale and have been rounded to one digit after the decimal. Estimate the relationship between GPA and ACT scores using OLS; that is, obtain beta_1 and beta_2 following equation: GPA_i = beta_0 + beta_1 ACT_i You may not use State or any other statistical software package to solve for and Comment on the direction of the relationship between ACT scores and GPA. Does the intercept have a useful interpretation here? Why do we still need to include an intercept term in this application? Explain. How much higher is GPA predicted to be if the ACT scores increase by 5 points? Compute the predicted values and residuals for each observation and add these values to the table Verify that the residuals sum (approximately) to zero and verify that the mean of the predicted values (approximately) equals the mean of Y (i.e., Y = Y): What is the predicted value of GPA when ACT = 25? How much of the variation in GPA for these eight students is explained by ACT? Explain.

Explanation / Answer

Following is the output of regression analysis:

(a)

The requried regression equation is

GPA= 0.5681 + 0.1022 ACT

Relationship between the variables is direct.

(b)

Intercept is 0.5681.

It says that if ACT =0 then GPA will be 0.5681. Yes it has meaning beucase ACT score can be zero.

(c)

For each unit increase in ACT , GPA is increased by 0.1022 units.

So 5 units increased in ACT GPA will increase by 5 * 0.1022 = 0.511 units.

(d)

Following is the completed table:

(e)

Sum of residuals are approximately zero.

(f)

Requried predicted value is 3.1231

(g)

R-square is : 0.577

That is 57.7% variation in GPA for these eight students is explained by ACT.

SUMMARY OUTPUT Regression Statistics Multiple R 0.759884064 R Square 0.577423791 Adjusted R Square 0.506994423 Standard Error 0.269173202 Observations 8 ANOVA df SS MS F Significance F Regression 1 0.594024725 0.594024725 8.19862235 0.0286766 Residual 6 0.434725275 0.072454212 Total 7 1.02875 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept 0.568131868 0.928421385 0.611933199 0.56303648 -1.703633417 2.839897154 ACT, X 0.102197802 0.035692019 2.863323654 0.0286766 0.014862578 0.189533026

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