Question 1

Which of the following is NOT a requirement for conducting inference on the least-squares regression model?

Accepted Answer

The explanatory variable $$x$$ must be normally distributed.

Question 2

Given the regression equation $$\hat{y} = 0.0116x + 5.5273$$, what is the predicted value of $$y$$ when $$x = 110$$?

Accepted Answer

$$6.8033$$

Question 3

What is the formula for the standard error of the estimate, $$s_e$$, in least-squares regression?

Accepted Answer

$$s_e = \sqrt{\dfrac{\sum (y_i - \hat{y}_i)^2$}{n-2}}$$

Question 4

A probability plot of residuals is used to check which assumption in regression analysis?

Accepted Answer

Normality of the residuals

Question 5

If the sum of squared residuals is $$2.7012$$ and the sample size is $$12$$, what is the standard error of the estimate?

Accepted Answer

$$0.5197$$

Question 6

Which of the following best describes the meaning of the slope $$\$$beta_1$$ in the regression model $$\mu_{y|x} = \$$beta_1 x$$ + \$$beta_0$$?

Accepted Answer

The average change in $$y$$ for a one-unit increase in $$x$$

Question 7

Which formula gives the test statistic for testing $$H_0$$: \$$beta_1 = 0$ in regression?

Accepted Answer

$$t_0$$ = \dfrac{$$b_1$$}{s_{$$b_1$$}}$$

Question 8

Suppose you conduct a two-tailed test for the slope and obtain $$t_0 = 3.867$ with 10$ degrees of freedom. If the critical values are $$\pm 2.228$$, what is the correct conclusion at $$\alpha = 0.05$$?

Accepted Answer

Reject the null hypothesis; there is evidence of a linear relationship.

Question 9

Which formula gives the lower bound for a $$(1-\alpha)\cdot100\%$$ confidence interval for the slope $$\$$beta_1$$?

Accepted Answer

$$b_1 - t$$_{\alpha/2} \cdot \dfrac{s_e}{\sqrt{\sum (x_i - \bar{x}$$)^2$$}}$$

Question 10

What is the main difference between a confidence interval for the mean response and a prediction interval for an individual response in regression?

Accepted Answer

A prediction interval is always wider than a confidence interval for the same $$x^*.$$

Question 11

Given $$\hat{y} = 6.8033$$, $$s_e = 0.5197$$, $$n = 12$$, $$x^* = 110$$, $$\bar{x} = 126.25$$, $$\sum (x_i - \bar{x})^2$ = 30006.25$$, and $$t_{0.025} = 2.228$$, what is the lower bound for the 95% confidence interval for the mean response at $$x^* = 110$$?

Accepted Answer

$$6.45$$

Question 12

Which of the following residual plots would indicate that the assumption of constant error variance is violated?

Accepted Answer

A plot where the spread of residuals increases as $$x$$ increases.

Question 13

If the P-value for a two-tailed test is $$0.003$$ and $$\alpha = 0.05$$, what is the correct decision?

Accepted Answer

Reject the null hypothesis.

Question 14

Which formula gives the upper bound for a 95% prediction interval for an individual response at $$x^*$$?

Accepted Answer

$$\hat{y} + t_{\alpha/2} \cdot s_e \sqrt{1 + \dfrac{1}{n} + \dfrac{(x^* - \bar{x})^2$}{\sum (x_i - \bar{x})^2$}}$$

Question 15

Why do we divide by $$n-2$$ instead of $$n$$ when calculating the standard error of the estimate in regression?

Accepted Answer

Because two parameters ($$\$$beta_0$$ and $$\$$beta_1) $$are estimated from the data.

Question 16

Which of the following best describes the critical region in a two-tailed $$t-$$test for the slope?

Accepted Answer

Values of the test statistic less than $$-t_{\alpha/2} or $$greater than $$t_{\alpha/2}$$

Question 17

In the context of regression, what does a large value of $$\sigma ($$the population standard deviation) indicate?

Accepted Answer

The data are widely dispersed about the regression line.

Question 18

Which of the following is a correct interpretation of a 95% confidence interval for the slope $$\$$beta_1$$?

Accepted Answer

We are 95% confident that the true mean change in $$y$$ for a one-unit increase in $$x is $$within the interval.

Inference on the Least-Squares Regression Model: Hypothesis Testing, Confidence Intervals, and Model Assumptions

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