Question 1

Which of the following best describes the coefficient of determination ($$R^2) in $$the context of a linear regression analysis?

Accepted Answer

It represents the percentage of variation in the response variable that is explained by the regression line.

Question 2

Suppose you have a dataset where the correlation between height in inches and height in centimeters is $$r = 1. $$What is the coefficient of determination ($$R^2) in $$this case, and what does it mean?

Accepted Answer

$$R^2 = 1$; all of the variation is perfectly explained by the regression line.

Question 3

A researcher is using the regression line to predict the value of a variable for an $$x-$$value far outside the range of the observed data. What is this practice called, and what is a potential risk?

Accepted Answer

Extrapolation; predictions may be unreliable because the relationship may not hold outside the observed range.

Question 4

In a scatterplot of height versus weight, the points are tightly clustered around the regression line. What does this indicate about the correlation and the coefficient of determination?

Accepted Answer

The correlation is strong (close to 1 or -1) and $$R^2 is $$high.

Question 5

Which of the following is an example of an influential point in regression analysis?

Accepted Answer

A data point that lies far from the regression line and significantly changes the slope if removed.

Question 6

If the probability that an event will happen is $$0.7$$, what is the probability that it will NOT happen?

Accepted Answer

$$0.3$$

Question 7

A box contains 10 red dice and 20 white dice. If one die is selected at random, what is the probability that it is red?

Accepted Answer

$$\frac{1}{3}$$

Question 8

In a survey, 30 people are married, 40 have a college degree, and 10 are both married and have a college degree. If a person is selected at random, what is the probability that the person is married OR has a college degree?

Accepted Answer

$$\frac{30 + 40 - 10}{100}$$

Question 9

Which of the following statements about mutually exclusive events is TRUE?

Accepted Answer

The probability that both occur is 0.

Question 10

A scatterplot shows a perfect horizontal line at $$y = 30. $$What is the coefficient of determination ($$R^2) $$for this data, and why?

Accepted Answer

$$R^2 = 0$; there is no variation to explain.

Question 11

If the correlation coefficient between two variables is $$r = -0.8$$, what is the coefficient of determination ($$R^2$$)?

Accepted Answer

$$0.64$$

Question 12

In a study of car values and age, the regression line explains about $$40\% of $$the variation in car value. What is the coefficient of determination ($$R^2$$)?

Accepted Answer

$$0.4$$

Question 13

Which of the following best describes the practical use of the coefficient of determination ($$R^2) in $$real-world data analysis?

Accepted Answer

It tells us the percentage of variation in the response variable explained by the predictor(s).

Question 14

If two events are complements, which of the following is always true?

Accepted Answer

Their probabilities add up to 1.

Question 15

A scatterplot of waist size versus weight shows a tighter clustering of points around the regression line than a scatterplot of height versus weight. What does this imply?

Accepted Answer

Waist size is a better predictor of weight than height.

Question 16

Which of the following is a correct interpretation of a coefficient of determination ($$R^2) of$$ $$0.85 in a $$regression analysis predicting blood pressure from age?

Accepted Answer

85% of the variation in blood pressure is explained by age.

Question 17

If the probability of drawing a heart from a standard deck of 52 cards is $$\frac{1}{4}$$, what is the probability of NOT drawing a heart?

Accepted Answer

$$\frac{3}{4}$$

Question 18

Which of the following is an example of extrapolation in regression analysis?

Accepted Answer

Using the regression line to predict a value for an $$x-$$value outside the observed data range.

Question 19

If the probability of event $$A is$$ $$0.4$$ and the probability of event $$B is$$ $$0.5$$, and $$A$$ and $$B$$ are mutually exclusive, what is the probability that either $$A or$$ $$B$$ occurs?

Accepted Answer

$$0.9$$

Regression, Correlation, and Probability in Statistics

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