Q: Given the regression equation $$\hat{y} = 545.290x - 827.355$$, what is the predicted y-value when $$x = 4$$?

$$\hat{y} = 545.290 \times 4 - 827.355 = 1353.805$$

Q: A researcher uses the regression equation $$\hat{y} = 1.709x + 79.101 to $$predict systolic blood pressure. What is the predicted blood pressure for a 50-year-old?

$$\hat{y} = 1.709 \times 50 + 79.101 = 164.551$$

Q: A city planner uses the regression equation $$\hat{y} = 0.054x + 14.13 to $$estimate the number of stories in a building based on its height in feet. What is the predicted number of stories for a building that is 600 feet tall?

$$\hat{y} = 0.054 \times 600 + 14.13 = 46.53$$

Question 1

Which of the following best describes the purpose of a regression line in statistics?

Accepted Answer

To model the relationship between two variables and predict y-values for given x-values.

Question 2

Given the regression equation $$\hat{y} = mx + b$$, what does the slope $$m$$ represent?

Accepted Answer

The change in the predicted y-value for a one-unit increase in x.

Question 3

Suppose you have the following summary statistics for a data set: $$n = 11$$, $$\sum x = 41$$, $$\sum y = 13,256$$, $$\sum xy = 64,776$$, $$\sum x^2$ = 181. $$What is the formula for the slope $$m of $$the regression line?

Accepted Answer

$$m = \frac{n \sum xy - (\sum x)(\sum y)}{n \sum x^2$ - (\sum x)^2$}$$

Question 4

What is a residual in the context of linear regression?

Accepted Answer

The difference between the observed y-value and the predicted y-value for a given x-value.

Question 5

Why does the regression line always pass through the point $$(\bar{x}, \bar{y})$$?

Accepted Answer

Because the regression line is constructed to ensure the means of x and y are on the line.

Question 6

Given the regression equation $$\hat{y} = 545.290x - 827.355$$, what is the predicted y-value when $$x = 4$$?

Accepted Answer

$$\hat{y} = 545.290 	imes 4 - 827.355 = 1353.805$$

Question 7

If the sum of the squares of the residuals is minimized for a line, what does this line represent?

Accepted Answer

The line of best fit (regression line).

Question 8

A researcher uses the regression equation $$\hat{y} = 1.709x + 79.101 to $$predict systolic blood pressure. What is the predicted blood pressure for a 50-year-old?

Accepted Answer

$$\hat{y} = 1.709 	imes 50 + 79.101 = 164.551$$

Question 9

Which of the following is NOT a valid step in constructing a regression line from a data set?

Accepted Answer

Draw a line that passes through all data points.

Question 10

A city planner uses the regression equation $$\hat{y} = 0.054x + 14.13 to $$estimate the number of stories in a building based on its height in feet. What is the predicted number of stories for a building that is 600 feet tall?

Accepted Answer

$$\hat{y} = 0.054 	imes 600 + 14.13 = 46.53$$

Question 11

Which of the following real-world scenarios is best modeled by a linear regression equation?

Accepted Answer

Predicting the number of calories burned based on minutes spent running.

Question 12

If the regression equation for a data set is $$\hat{y} = 2x + 5$$, what is the residual for the data point $$(x = 3, y = 15)$$?

Accepted Answer

Residual $$= 15 - (2 	imes 3 + 5) = 15 - 11 = 4$$

Question 13

Which formula is used to calculate the y-intercept $$b of $$the regression line?

Accepted Answer

$$b = \bar{y} - m\bar{x}$$

Question 14

A teacher uses the regression equation $$\hat{y} = 10x + 50 to $$predict test scores based on hours studied. If a student studies for 6 hours, what is the predicted test score?

Accepted Answer

$$\hat{y} = 10 	imes 6 + 50 = 110$$

Linear Regression: Correlation and Regression in Statistics

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