Question

Use the data on age of fetuses and length of crown-rump from Exercise \(14.26\)

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.

Short answer

Part a. The standard error of the estimate is \(5.52\)

Part b. The variables age of fetuses \((x)\) and the length of crown-rump \((y)\) satisfy the assumptions for regression inferences,

Part c.

Part d. The assumptions for regression inferences are satisfied.

Step-by-step solution

Part a. Step 1. Given information

Given,

Part a. Step 2. Calculation

Analysis of Variance:

From the above Analysis of variance printout,

We have \(SSE = 244\),

\(df=n-2\)

\(= 10-2\)

\(=8\)

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

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

\(=\sqrt{\frac{244}{8}}\)

\(=\sqrt{30.5}\)

\(=5.522680509\)

\(\approx 5.52\)

Interpretation: Very roughly speaking, on average, the predicted length of crown-rump in the sample differs from the observed length of crown-rump by about \(552\) millimeters.

Therefore, the standard error of the estimate is \(5.52\).

Part b. Step 1. Calculation

Interpretation:

Presuming that, for crown-rump, the variables age of fetuses (x) and the length of crown-rump (y) satisfy the assumptions for regression inferences, the standard error of the estimate, provides an estimate for the common population standard deviation, , of length of crown-rump, in millimeter's of any particular age of fetuses (weeks).

Therefore, the variables age of fetuses \((x)\) and the length of crown-rump \((y)\) satisfy the assumptions for regression inferences.

Part c. Step 1. Calculation

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.

Part d. Step 1. Calculation

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.