Use the data on total hours studied over \(2\) weeks and test score at the end of the \(2\) weeks from Exercise \(14.27\).

a. compute the standard error of the estimate and interpret your answer.

b. interpret your results from part (a) if the assumptions for regression inferences hold.

c. obtain a residual plot and a normal probability plot of the residuals.

d. decide whether you can reasonably consider Assumptions \(1-3\) for regression inferences to be met by the variables under considerations.

Given,

Analysis of Variance:

From the above Analysis of variance printout,

We have \(SSE = 75.11\),

\(df=n-2\)

\(= 8-2\)

\(=6\)

The formula for the standard error of the estimate is given by.

\(s_{e}=\sqrt{\frac{SSE}{n-2}}\)

\(=\sqrt{\frac{75.11}{6}}\)

\(=\sqrt{12.5183333}\)

\(=3.538125681\)

\(\approx 3.54\)

Interpretation: Very roughly speaking. on average, the predicted test score of a student in the sample differs from the observed score by about \(354\) points.

Therefore, the standard error of the estimate is \(3.54\).

Interpretation:

Presuming that, for students in beginning calculus courses, the variables study time \((x)\) and test score \((y)\) satisfy the assumptions for regression inferences, the standard error of the estimate, \(s_{e}=3.54\) provides an estimate for the common population standard deviation, \(\sigma\) of test scores for all students who study for any particular amount of time.

Therefore, the variables study time \((x)\) and test score \((y)\) satisfy the assumptions for regression inferences.

Residual plot:

MINITAB procedure:

Step 1: Choose Stat > Regression > Regression.

Step 2: In Response, enter the column \(y\)

Step 3: In Predictors, enter the columns \(x\).

Step 4: In Graphs, enter the columns x variables under Residuals versus the variables.

Step 5: Click OK.

MINITAB output:

Normal probability plot:

MINITAB procedure:

Step 1: Choose Stat > Regression > Regression.

Step 2: In Response, enter the column \(y\)

Step 3: In Predictors, enter the columns \(x\).

Step 4: In Graphs, select Normal probability plot of residuals.

Step 5: Click OK.

MINITAB output:

Therefore, residual plot and normal probability plot are constructed.

The above plots in part (c) suggest that the assumptions \(1-3\) for regression inferences appear to be met, but the residual plot suggest that the assumptions \(1\) and \(2\) may be violated. It is reasonable to believe that the assumptions for regression inferences are satisfied.

Therefore, the assumptions for regression inferences are satisfied.