Question 1

Which of the following best describes the relationship shown in a scatter diagram where, as the predictor variable increases, the response variable also increases in a roughly straight-line pattern?

Accepted Answer

Positive linear association

Question 2

Given the regression equation $$\hat{y} = 5.8 + 2.4x$$, what is the predicted value of $$y$$ when $$x = 7$$?

Accepted Answer

$$\hat{y} = 22.6$$

Question 3

Which formula is used to calculate the sample correlation coefficient $$r$$ between two variables $$x$$ and $$y$$?

Accepted Answer

$$r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{(n-1)s_x s_y}$$

Question 4

If the correlation coefficient $$r$$ for a dataset is $$-0.85$$, which of the following best describes the association between the two variables?

Accepted Answer

Strong negative linear association

Question 5

Which of the following statements about the least-squares regression line is TRUE?

Accepted Answer

It minimizes the sum of the squared vertical distances between observed and predicted values.

Question 6

Suppose the coefficient of determination $$R^2$$ for a regression model is $$0.80. $$What does this mean?

Accepted Answer

80% of the variation in the response variable is explained by the predictor variable.

Question 7

Which of the following is NOT an assumption required for hypothesis testing of the slope coefficient in simple linear regression?

Accepted Answer

The predictor variable must be categorical.

Question 8

A regression analysis yields $$r = 0.96. $$What is the value of $$R^2$$ and how should it be interpreted?

Accepted Answer

$$R^2 = 0.9216$, meaning 92.16% of the variation in y$ is explained by x$.

Question 9

Which of the following best describes the meaning of the error term $$u in $$the regression model $$y = b_0$ + b_1$ x + u$$?

Accepted Answer

It represents the difference between the observed and predicted values of $$y.$$

Question 10

In the context of regression, what does it mean if the sum of the residuals $$\sum u_i is $$zero?

Accepted Answer

The regression line passes through the mean of $$x$$ and $$y.$$

Question 11

Which formula is used to calculate the slope $$b_1 of $$the least-squares regression line?

Accepted Answer

$$b_1$$ = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x}$$)^2$$}$$

Question 12

A researcher wants to test if there is a significant linear relationship between two variables using a two-tailed test at $$\alpha = 0.05$$ with $$n = 5$$ observations. What is the critical $$t-$$value?

Accepted Answer

$$\pm 3.182$$

Question 13

Which of the following is a correct interpretation of the $$y-$$intercept $$b_0 in a $$regression equation?

Accepted Answer

It is the predicted value of $$y$$ when $$x = 0$$, provided $$x = 0 is $$within the scope of the data.

Question 14

Which of the following best describes extrapolation in the context of regression analysis?

Accepted Answer

Predicting values of the response variable for predictor values outside the observed data range.

Question 15

If the standard error of the estimate $$s_e is $$large, what does this indicate about the regression model?

Accepted Answer

The observed values are widely scattered around the regression line.

Question 16

Which of the following is the correct formula for the standard error of the estimate $$s_e$$?

Accepted Answer

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

Question 17

Which of the following best describes the coefficient of determination $$R^2$$?

Accepted Answer

It measures the proportion of total variation in $$y$$ explained by the regression line.

Question 18

In hypothesis testing for the slope $$\$$beta_1$$ in regression, which of the following is the null hypothesis?

Accepted Answer

$$H_0$$: \$$beta_1 = 0$

Question 19

Which of the following Excel tools would you use to calculate the correlation coefficient between two variables?

Accepted Answer

Data Analysis Toolpak: Correlation

Describing the Relation Between Two Variables: Scatter Diagrams, Correlation, and Regression

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