Q: A regression analysis yields the least squares regression line $$\hat{y} = 2 + 3x. $$What is the predicted value of $$y$$ when $$x = 4$$?

$$\hat{y} = 2 + 3 \times 4 = 14$$

Q: Suppose the mean and standard deviation of $$x$$ are $$\bar{x} = 10$$ and $$s_x = 2$$, and for $$y$$ are $$\bar{y} = 50$$ and $$s_y = 10. $$The correlation between $$x$$ and $$y is$$ $$r = 0.6. $$What is the slope $$b of $$the least squares regression line?

$$b = r \frac{s_y}{s_x} = 0.6 \times \frac{10}{2} = 3$$

Question 1

A medical researcher is studying the relationship between smoking status (smoker/non-smoker) and the presence of a certain lung disease (present/absent) in a sample of 500 adults. Which of the following would be the most appropriate way to display the relationship between these two categorical variables?

Accepted Answer

A segmented bar graph with smoking status on the x-axis and relative frequency of disease presence on the y-axis.

Question 2

In a study of exercise and cholesterol level, the correlation coefficient $$r$$ between hours of exercise per week and cholesterol level (mg/dL) is found to be $$-0.85. $$Which of the following is the best interpretation of this value?

Accepted Answer

There is a strong negative linear association between hours of exercise and cholesterol level.

Question 3

A two-way table summarizes the relationship between gender (male/female) and preference for a new product (like/dislike). The conditional relative frequency of 'like' among females is 0.60, and among males is 0.40. What is the relative risk of females liking the product compared to males?

Accepted Answer

$$\frac{0.60}{0.40} = 1.5$$

Question 4

Which of the following statements about the correlation coefficient $$r is $$TRUE?

Accepted Answer

$$r is $$affected by outliers and is not resistant.

Question 5

A regression analysis yields the least squares regression line $$\hat{y} = 2 + 3x. $$What is the predicted value of $$y$$ when $$x = 4$$?

Accepted Answer

$$\hat{y} = 2 + 3 	imes 4 = 14$$

Question 6

A scatterplot of the number of hours studied (x) and exam score (y) for 30 students shows a strong, positive, linear association. Which of the following best describes the form, direction, and strength of this association?

Accepted Answer

Linear, positive, strong

Question 7

Suppose the mean and standard deviation of $$x$$ are $$\bar{x} = 10$$ and $$s_x = 2$$, and for $$y$$ are $$\bar{y} = 50$$ and $$s_y = 10. $$The correlation between $$x$$ and $$y is$$ $$r = 0.6. $$What is the slope $$b of $$the least squares regression line?

Accepted Answer

$$b = r \frac{s_y}{s_x} = 0.6 	imes \frac{10}{2} = 3$$

Question 8

A residual plot for a linear regression model shows a clear curved pattern. What does this indicate about the appropriateness of the linear model?

Accepted Answer

The linear model is not appropriate; a non-linear model may be better.

Question 9

Which of the following best describes the meaning of the coefficient of determination $$r^2 in $$the context of a regression analysis?

Accepted Answer

$$r^2 is $$the proportion of variation in the response variable explained by the regression line.

Question 10

A researcher fits a regression line to data on age (years) and blood pressure (mmHg) and finds the y-intercept is $$a = 80. $$Which of the following is a correct interpretation of the y-intercept in this context?

Accepted Answer

It is the predicted blood pressure for a person who is 0 years old.

Question 11

Which of the following is NOT a property of the correlation coefficient $$r$$?

Accepted Answer

$$r is $$resistant to outliers.

Question 12

A point in a scatterplot has an $$x-$$value much larger than the rest of the data, but its $$y-$$value fits the general trend. This point is best described as:

Accepted Answer

A high-leverage point

Question 13

In a study, the sum of the residuals from the least squares regression line is calculated. What is the value of this sum?

Accepted Answer

0

Question 14

Which of the following scenarios is an example of extrapolation?

Accepted Answer

Using a regression line based on data for ages 20-60 to predict the value for age 80.

Question 15

A segmented bar graph is used to compare two categorical variables. Which variable should be placed on the x-axis?

Accepted Answer

The explanatory variable

Question 16

Which of the following is a correct interpretation of the slope $$b in $$the regression equation $$\hat{y} = a + bx$$?

Accepted Answer

The predicted change in $$y$$ for every one unit increase in $$x.$$

Question 17

Which of the following would indicate that two categorical variables are independent when displayed in a segmented bar graph?

Accepted Answer

The bars have the same or very close percents.

Question 18

A data set contains an observation that, if removed, would substantially change the slope and y-intercept of the regression line. This observation is best described as:

Accepted Answer

An influential point

Question 19

If the correlation between two variables is $$r = 0.95$$, but the scatterplot shows a clear curved pattern, which of the following is TRUE?

Accepted Answer

The correlation is high, but a linear model is not appropriate.

Question 20

Which of the following transformations might be used if a scatterplot of $$x$$ and $$y$$ shows a non-linear relationship, and a more linear relationship is desired?

Accepted Answer

Taking the natural log of $$y$$

Exploring Two-Variable Data: Categorical and Quantitative Relationships

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