Question 1

Which of the following best describes the role of a residual plot in regression analysis?

Accepted Answer

It helps determine if the linear model is appropriate by revealing patterns or bends not visible in the original scatterplot.

Question 2

Suppose you are analyzing the relationship between the number of hours studied and exam scores for a group of students. You calculate a correlation coefficient $$r = 0.85. $$Which of the following is the best interpretation?

Accepted Answer

There is a strong positive linear relationship between hours studied and exam scores.

Question 3

A researcher fits a linear regression model to predict fuel efficiency (mpg) from car weight (kg). The regression equation is $$\hat{y} = 40 - 0.01x. $$What does the slope of this equation mean in context?

Accepted Answer

For every 1 kg increase in car weight, fuel efficiency decreases by 0.01 mpg.

Question 4

Which of the following conditions must be checked before interpreting the correlation coefficient $$r$$?

Accepted Answer

Quantitative Variables Condition, Straight Enough Condition, and No Outliers Condition

Question 5

Given the formula for the slope of the least squares regression line $$b_1 = r$$ \cdot \frac{s_y}{s_x}$$, what would happen to the slope if the standard deviation of x$ (s_x$) increases while r$ and s_y$ remain constant?

Accepted Answer

The slope $$b_1$$ would decrease.

Question 6

A scatterplot of average life expectancy versus number of TVs per person in a country shows a strong positive association. Which of the following is the best conclusion?

Accepted Answer

There is an association, but a lurking variable such as standard of living may explain the relationship.

Question 7

Which of the following statements about $$R^2 ($$the coefficient of determination) is correct?

Accepted Answer

$$R^2 is $$the square of the correlation coefficient and represents the proportion of variability in $$y$$ explained by $$x.$$

Question 8

A student fits a linear model to data and finds a residual plot with a clear curve. What should the student conclude?

Accepted Answer

The data violate the Straight Enough Condition, and a nonlinear model may be more appropriate.

Question 9

Which of the following is a potential danger of extrapolation when using a linear regression model?

Accepted Answer

Predictions may be unreliable because the relationship between $$x$$ and $$y$$ may change outside the observed range.

Question 10

In a regression analysis, what is the effect of a data point with high leverage?

Accepted Answer

It can pull the regression line closer to itself, potentially changing the slope and intercept significantly.

Question 11

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

Accepted Answer

It is affected by changes in the center or scale of the variables.

Question 12

A regression model is used to predict hurricane wind speed from central pressure. The standard error of the residuals ($$s_e) is 16.2 $$knots. What does this mean?

Accepted Answer

On average, the model's predictions are off by about 16.2 knots.

Question 13

Which of the following best describes the difference between association and correlation?

Accepted Answer

Association is a general relationship between variables, while correlation is a specific, calculated measure for linear relationships.

Question 14

If you switch the explanatory and response variables in a regression analysis, what happens to the regression line?

Accepted Answer

The regression line changes, unlike the correlation coefficient which remains the same.

Question 15

Why do scatterplots of summary statistics (such as state averages) tend to show less variability than scatterplots of individual data points?

Accepted Answer

Summary statistics vary less than individual values, which can give a false impression of a stronger relationship.

Question 16

Which of the following is a correct interpretation of a negative residual in a regression analysis?

Accepted Answer

The model overestimated the observed value.

Question 17

A regression analysis of shoe size and IQ includes a single data point with an extremely large shoe size. What is the likely effect of this point?

Accepted Answer

It has high leverage and can significantly alter the regression line.

Question 18

Which of the following is a lurking variable in the context of the relationship between number of doctors per person and life expectancy?

Accepted Answer

Standard of living, which affects both the number of doctors and life expectancy.

Question 19

Which of the following is NOT a recommended practice when dealing with unusual points in regression analysis?

Accepted Answer

Simply delete unusual points to improve the model fit without further investigation.

Scatterplots, Correlation, and Linear Regression: Study Guide for Statistics Students

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