Question 1

Which of the following best describes the purpose of plotting residuals after fitting a regression model?

Accepted Answer

To reveal patterns or unusual aspects in the data that may not be visible in the initial scatter plot.

Question 2

A residual plot that shows a random scatter of points around zero suggests which of the following?

Accepted Answer

The regression model is a good fit for the data.

Question 3

Given the regression equation $$\hat{y} = 284x - 1601$$ and $$x_0 = 86$, what is the predicted value $$\hat{y}_0$$?

Accepted Answer

$$\hat{y}_0 = 8323$$

Question 4

Which formula is used to calculate the margin of error $$E$$ for a prediction interval in linear regression?

Accepted Answer

$$E = t_{\alpha/2} \cdot s_e \sqrt{1 + \frac{1}{n} + \frac{n(x_0$ - \bar{x})^2$}{n\sum x^2$ - (\sum x)^2$}}$$

Question 5

Why is it risky to use a regression model to make predictions for $$x$$ values outside the range of the observed data?

Accepted Answer

Because extrapolation can lead to unreliable predictions that are not supported by the data.

Question 6

In the context of regression analysis, what does a residual represent?

Accepted Answer

The difference between the observed $$y$$ value and the predicted $$y$$ value.

Question 7

A residual plot for a regression model shows a clear curve pattern. What does this indicate?

Accepted Answer

There is likely a non-linear relationship between the variables.

Question 8

When checking for subgroups in data (such as shelf placement in the cereal example), what is the main reason for doing so?

Accepted Answer

To identify if different groups have different relationships between $$x$$ and $$y.$$

Question 9

Which of the following steps is NOT part of making a prediction interval using a TI-84 calculator?

Accepted Answer

Set the regression slope to zero.

Question 10

If the 95% prediction interval for ice cream sales at 86°F is ($$6054.7$$, $$10591.3$$), what does this interval mean?

Accepted Answer

We are 95% confident that a single day's sales at 86°F will fall between $$6054.7$$ and $$10591.3.$$

Question 11

Which of the following is a correct interpretation of a residual plot that shows a funnel shape (wider spread as $$x$$ increases)?

Accepted Answer

There is evidence of heteroscedasticity in the data.

Question 12

What is the correct formula for the degrees of freedom used when finding the critical value $$t_{\alpha/2}$$ for a prediction interval in simple linear regression?

Accepted Answer

$$df = n - 2$$

Question 13

Which of the following is a common mistake when interpreting residual plots?

Accepted Answer

Ignoring patterns in the residuals that suggest model inadequacy.

Question 14

If you want to predict $$y$$ for a new $$x$$ value using a regression model, which of the following must be true for the prediction to be reliable?

Accepted Answer

The $$x$$ value should be within the range of the observed data.

Question 15

Which of the following best describes the margin of error $$E in a $$prediction interval for regression?

Accepted Answer

It is the amount added and subtracted from the predicted $$y to $$form the prediction interval.

Question 16

Which of the following is NOT a reason to examine residuals after fitting a regression model?

Accepted Answer

To increase the value of the correlation coefficient.

Residuals, Regression, and Prediction Intervals in Business Statistics

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