Question

In Exercises 14.34-14.43, use the technology of your choice to
a. obtain and interpret the standard error of the estimate.
b. obtain a residual plot and a normal probability plot of the residuals.
c. decide whether you can reasonably consider Assumptions I-3 for regression inferences met by the two variables under consideration.

14.42 Estriol Level and Birth Weight. J. Greene and J. Touchstone conducted a study on the relationship between the estriol levels of pregnant women and the birth weights of their children. Their findings, "Urinary Tract Estriol: An Index of Placental Function," were published in the American Journal of Obstetrics and Gynecology (Vol. 85(1), pp. 1-9). The data points are provided on the WeissStats site, where estriol levels are in $\mathrm{mg}/24\mathrm{hr}$ and birth weights are in hectograms $\left(hg\right)$.

Short answer

(a) The standard error of estimate is $3.821$.

(b) The points closely resemble the normal probability plot that obtained from the residual plot and the normal probability plot of the residuals.

(c) There is no violation of regression inferences assumptions 1-3, and it is appropriate to consider the regression inferences assumptions met for the variables under examination, for the provided sample.

Step-by-step solution

Part (a) Step 1: Given information

To obtain the standard error of the estimate and interpret.

Part (a) Step 2: Explanation

A sample of residences in a certain area's lot sizes and assessed values are provided.
The Residual as follows:

The error is described as $e=\hat{y}-y$, where $\hat{y}$is the projected response variable value and $y$ is the actual response variable value.
The standard error of estimate is calculate as follows:

The standard error of estimation for a collection of $n$observations is given as:

$S_e=\sqrt{\frac{SSE}{n-2}}$

where SSE stands for squared error sum
Then the Residual plot will be computed as follows:

As a result, the standard error of estimate is $3.821$.
The predicted birth weights deviate by $3.821hg$on average from the observed birth weights, which is a point estimate of common standard deviation.

Part (b) Step 1: Given information

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

Part (b) Step 2: Explanation

Obtain the residual plot of the residuals by using MINITAB as follows:

The output will be:

The residuals, or values of $e$, corresponding to the values of $x$ are plotted on the graph .
The normal probability plot of the residuals by using MINITAB as follows:

The points closely resemble the normal probability plot.

Part (c) Step 1: Given information

To consider Assumptions$1-3$ for regression inferences met by the two variables under consideration.

Part (c) Step 2: Explanation

Let, assumption for regression inferences as follows:

• Population regression line: There are constants ${\beta }_0$and ${\beta }_1$such that the conditional mean of the response variable $\left(y\right)$is ${\beta }_0+{\beta }_{1}x$for each value of the predictor variable $\left(x\right)$.
• Equal standard deviation: The response variable $\left(y\right)$has the same conditional standard deviation $\sigma$for all values of the predictor variable$\left(x\right)$.
• For any value of the predictor variable $\left(x\right)$, the conditional distribution of the response variable $\left(y\right)$is a normal distribution.
• Independent observations: The responses variable's observations are independent to one another.

Let, the assumption for residual analysis for the regression model is considered as follows:

• The residuals should fall roughly in a horizontal band centered and symmetric about the $x$-axis when plotted against the recorded values of the predictor variable.
• The residuals in a normal probability plot should be nearly linear.

The normal probability plot is linear, indicating that the assumption of normality is met. The residuals plot shows that there is increasing variability and that the $x$ values vary.
As a result, there is only a minor divergence from the equality of variance, although this is easily overlooked.
As a result, there is no violation of regression inferences assumptions $1-3$, and it is appropriate to consider the regression inferences assumptions met for the variables under examination, for the provided sample