Question

Margin of Error in Regression. In Exercises14.109and14.110, you will examine the magnitude of the margin of error of confidence intervals and prediction intervals in regression as a function of how far the specified value of the predictor variable is from the mean of the observed values of the predictor variable.
14.109 Age and Price of Orions. Refer to the data on age and price of a sample of 11 Orions given in Table 14.4 on page $572$
a. For each age between 2 and 7 years, obtain a95%confidence interval for the mean price of all Orions of that age. Plot the confidence intervals against age and discuss your results.
b. Determine the margin of error for each confidence interval that you obtained in part (a). Plot the margins of error against age and discuss your results.
c. Repeat parts (a) and (b) for prediction intervals.

Short answer

Part a)

Look at the figure confidence intervals are shaded by yellow color

The regression equation is not fitted well, because only one point is not within the bands of the confidence interval.

Part b)

There is a non-linear relationship found between the margin of error and age

Part c)

Repeat part (a)

Look at the figure, the regression equation is fitted well because all ages are within the bands of the prediction interval.

Repeat part b)

There is a nonlinear relationship found between margin of error and age.

Step-by-step solution

Part a) Step 1: Explanation

Let $x$is the age in years.

Let $y$is the price in$\$100$

Obtain a 95\% confidence interval for the mean price of all Orions of that age.

Using Minitab follow the steps for finding the $95\%$confidence intervals.

1. Enter data into Minitab Worksheet.

2. Click Minitab Enable commands for getting commands used in finding confidence intervals through regression.

3. The obtained Minitab partial output for the $95\%$confidence intervals is,

Figure 1

Look at the figure $95\%$confidence intervals are shaded by yellow color.

Part a) Step 2: Explanation

Plot the confidence intervals against age.

Minitab commands:

The obtained output for the confidence intervals plot is,

Figure (2)

Look at the figure, the regression equation is not fitted well, because only one point is not within the bands of the confidence interval.

Part b) Step 1: Explanation

Find the margin of error for each $95\%$confidence interval obtained in part (a).

The formula for the margin of error$E$is,

$E=\frac{upperlimit-lowerlimit}{2}$

Using this formula in Minitab, the output for the margin of errors is;

Using the formula of Minitab the output of the margin error is

The scatterplot for the margin of error against age is

There is a non-linear relationship found between the margin of error and age

Part c) Step 1: Explanation

Obtain a $95\%$prediction interval for the mean price of all Orions of that age.

Using Minitab follow the steps for finding the $95\%$confidence intervals.

1. Enter data into Minitab Worksheet.

2. Click Minitab Enable commands for getting commands used in finding confidence intervals through regression.

3. The obtained Minitab partial output for the $95\%$confidence intervals is,

Look at the figure, $95\%$prediction intervals are shaded by yellow color.

Plot the prediction intervals against age.

The obtained output for the prediction intervals plot is,

Figure (2)

Look at the figure, the regression equation is fitted well because all ages are within the bands of the prediction interval.

Part c) Step 2: Explanation

Repeat part (b)

Find the margin of error for each $95\%$confidence interval obtained in part (a).

The formula for the margin of error$E$is,

$E=\frac{upperlimit-lowerlimit}{2}$

Minitab Command:

Using this formula in Minitab, the output for the margin of errors is,

The scatterplot for the margin of errors against age is,

Figure (4)

There is a nonlinear relationship found between the margin of error and age.