Facts about the Least-Squares Regression

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Facts about least-squares Line

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Questions 1-3

Question 1.

True or false: Both correlation and linear regression require a straight line relationship between X and Y.

False

True

Correct. This is a correct statement.

Incorrect. This is a correct statement.

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Question 3.

What does regression give us?

A measure of direction and strength of the linear relationship between x and y

A model of the relationship between x and y

Correct. Regression gives us a formula for the line:

\hat{y} = a + b x

$\widehat{y} = a + bx$ .

Incorrect. Regression gives us a formula for the line:

\hat{y} = a + b x

$\widehat{y} = a + bx$ .

Try again.

Questions 4-6

Question 4.

True or false: X and Y can be interchanged in both correlation and linear regression.

True

False

Correct. X and Y can be interchanged in correlation, but not in the formula for a regression line:

\hat{y} = a + b x

$\widehat{y} = a + bx$ .

Incorrect. X and Y can be interchanged in correlation, but not in the formula for a regression line:

\hat{y} = a + b x

$\widehat{y} = a + bx$ .

Try again.

Question 5.

Which of the following requires an important distinction between X and Y?

Correlation

Regression

Correct. X and Y can be interchanged in correlation, but not in the formula for a regression line:

\hat{y} = a + b x

$\widehat{y} = a + bx$ .

Incorrect. X and Y can be interchanged in correlation, but not in the formula for a regression line:

\hat{y} = a + b x

$\widehat{y} = a + bx$ .

Try again.

Question 6.

Which of the following makes no distinction as to which variable is X and which is Y?

Regression

Correlation

Correct. X and Y can be interchanged in correlation, but not in the formula for a regression line:

\hat{y} = a + b x

$\widehat{y} = a + bx$ .

Incorrect. X and Y can be interchanged in correlation, but not in the formula for a regression line:

\hat{y} = a + b x

$\widehat{y} = a + bx$ .

Try again.

Question 12

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Question 12.

True or false: The regression line always passes through the point ( $\overline{x}$ , $\overline{y}$ ).

False

True

Correct. This is a correct statement.

Incorrect. This is a correct statement.

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Question 17

502

Question 17.

True or false: r² is a measure of how successfully the regression line explains the variation in y.

True

False

Correct. This is a correct statement.

Incorrect. This is a correct statement.

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Questions 18-19

593

Question 18.

True or false: If r² is really close to 100%, then the sum of squared residuals is very small.

True

False

Correct. This is a correct statement.

Incorrect. This is a correct statement.

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Question 20

631

Question 20.

True or false: r² measures the fraction of y values that are exactly predicted by the x values.

False

True

Correct. r² is a measure of the fraction of variation in the y’s that is explained by x. It does not tell us the fraction of y values that are exactly predicted as most are not, even when r² is close to 100%.

Incorrect. r² is a measure of the fraction of variation in the y’s that is explained by x. It does not tell us the fraction of y values that are exactly predicted as most are not, even when r² is close to 100%.

Try again.

Questions 21-22

754

Question 21.

What does total variation in the y’s measure?

The variability of the y’s about their mean,

\bar{y}

$\overline{y}$ .

The variability of the y’s about the regression line.

Correct. The variability of the y’s about their mean

\bar{y}

$\overline{y}$ is called the total variation in y. The variability of the y’s about the regression line is a measure of the prediction errors (or residuals).

Incorrect. The variability of the y’s about their mean

\bar{y}

$\overline{y}$ is called the total variation in y. The variability of the y’s about the regression line is a measure of the prediction errors (or residuals).

Try again.

Questions 23-24

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Question 23.

Suppose two different explanatory variables have a linear relationship with a response variable, y. The regression line using variable #1 has an r² of 47% and the regression line using variable #2 has an r² of 89%. Which explanatory variable explains the most variation in y?

Variable #1

Variable #2

Correct. r² tells us the percentage of variation in y that is explained by x. And the closer r² is to 100%, the greater the variation in y that gets explained. So an r² of 89% tells us that more variation in y is explained than when r² is 47%.

Incorrect. r² tells us the percentage of variation in y that is explained by x. And the closer r² is to 100%, the greater the variation in y that gets explained. So an r² of 89% tells us that more variation in y is explained than when r² is 47%.

Try again.

StatTutor Lesson - Facts about the Least-Squares Regression

Questions 1-3

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Question 2.

Question 3.

Questions 4-6

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Questions 7-11

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Question 12

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Questions 13-16

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Question 17

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Questions 18-19

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Question 20

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Questions 21-22

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Questions 23-24

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Question 24.