Linear Regression

Show your work for each problem.

1.
What does the slope in a regression line represent?
1. a)The average of all y values
2. b)The number of data points
3. c)How much y changes for each unit increase in x
4. d)The value of y when x = 0
Answer: How much y changes for each unit increase in x
The slope represents how much changes for each one-unit increase in . It's the rate of change of the relationship.
2.
In the regression line , what is the y-intercept?
1. a)10
2. b)3
3. c)13
4. d)30
Answer: 10
In , the y-intercept is 10. This is the value of when : .
3.
Using , what is the predicted value of when ?
Answer: 13
4.
If a regression line has slope and passes through the point , what is the y-intercept?
Answer: 7
The point means when , . This is exactly the definition of the y-intercept, so .
5.
A study finds that predicts test scores () based on study hours (). What does the 5 mean?
1. a)The base score with no studying
2. b)Each additional hour of study adds 5 points
3. c)The maximum possible score
4. d)The number of students in the study
Answer: Each additional hour of study adds 5 points
The slope 5 means that for each additional hour of study, the predicted test score increases by 5 points. The 20 is the base score (y-intercept).
6.
Using , predict when .
Answer: 40
7.
What is a residual in linear regression?
1. a)The difference between actual and predicted values
2. b)The y-intercept of the line
3. c)The slope of the regression line
4. d)The average of all x values
Answer: The difference between actual and predicted values
A residual is . It shows how much the actual data point differs from what the regression line predicts.
8.
Find the y-intercept of a regression line with slope that passes through .
Answer: 5
1. Write the equation form y = mx + b
2. Substitute m = 3, x = 2, y = 11 11 = 3(2) + b
3. Simplify 3(2) 6
4. Solve for b: 11 = 6 + b 5
9.
The actual value is and the predicted value is . What is the residual?
Answer: 3
Residual = . A positive residual means the actual value is above the regression line.
10.
Given , , and slope , find the y-intercept.
Answer: 10
1. Write the formula for b b = y-bar - m * x-bar
2. Calculate m times x-bar: 4 × 10 40
3. Subtract from y-bar: 50 - 40 10
11.
Why should you be cautious about using regression to predict values far outside your data range?
1. a)The y-intercept is undefined outside the range
2. b)The calculations become too difficult
3. c)The slope might change outside the data range
4. d)The relationship may not hold beyond the observed data
Answer: The relationship may not hold beyond the observed data
Extrapolating beyond your data assumes the linear relationship continues indefinitely, but in reality, relationships often change. A linear trend in one range may become curved or reverse outside that range.
12.
Given data: . Find the slope of the regression line.
Answer: 2
1. Calculate x-bar (mean of x values: 1, 2, 3) 2
2. Calculate y-bar (mean of y values: 3, 5, 7) 5
3. Calculate numerator: sum of (x-x̄)(y-ȳ) 4
4. Calculate denominator: sum of (x-x̄)² 2
5. Divide to get slope m = 4/2 2
13.
A regression line is . If , what is the predicted ?
Answer: 70
14.
If the sum of squared residuals for a line is minimized, what does this mean?
1. a)The line passes through all points
2. b)All residuals are zero
3. c)This is the line of best fit
4. d)The slope is zero
Answer: This is the line of best fit
The line that minimizes the sum of squared residuals is the line of best fit. This is the definition of the least squares regression line.
15.
For the data with slope , find the complete regression equation.
Answer: y = 2x + 1
1. Find x-bar 2
2. Find y-bar 5
3. Use b = y-bar - m·x-bar. Calculate m·x-bar 4
4. Calculate b = 5 - 4 1
5. Write the equation y = mx + b y = 2x + 1
16.
Which of these best describes the purpose of the line of best fit?
1. a)To show the general trend in the data
2. b)To connect the first and last points
3. c)To pass through every data point
4. d)To maximize the distance from points
Answer: To show the general trend in the data
The line of best fit shows the general trend or pattern in the data, not individual points. It helps us understand the overall relationship between variables.