Introduction to Regression

Show your work for each problem.

1.
What does the represent in the equation ?
1. a)The x-value
2. b)The slope (rate of change)
3. c)The y-intercept
4. d)The prediction
Answer: The slope (rate of change)
In the equation , represents the slope, which tells us how much changes for every 1 unit increase in .
2.
What is the purpose of linear regression?
1. a)To calculate the average of numbers
2. b)To find the best-fitting line through data points
3. c)To count how many data points exist
4. d)To find the largest value in a dataset
Answer: To find the best-fitting line through data points
Linear regression finds the straight line that best fits a set of data points, allowing us to make predictions and understand relationships between variables.
3.
For the equation , what is the slope?
Answer: 3
In , the slope is . This is the coefficient of , meaning increases by 3 for every 1 unit increase in .
4.
For the equation , what is the y-intercept?
Answer: 5
In , the y-intercept is . This is where the line crosses the y-axis (when ).
5.
Using the equation , find when .
Answer: 14
. We substituted into the equation and simplified.
6.
Calculate the slope between points and .
1. a)8
2. b)4
3. c)2
4. d)0.5
Answer: 2
Slope . The y-values change by 8 while x-values change by 4.
7.
Calculate the slope between points and .
Answer: 3
Slope . For every increase of 1 in x, y increases by 3.
8.
Find the equation of the line through points and .
Answer: y = 3x
1. Calculate the slope using the formula . What is ? 3
2. Use point and in . Solve for : . What is ? 0
3. Write the final equation with and . y = 3x
9.
A regression model shows that test scores increase by 8 points for each hour of studying. A student who studies 0 hours scores 40 points. Write the equation and predict the score for 5 hours of studying.
Answer: 80
1. What is the slope (points gained per hour)? 8
2. What is the y-intercept (score at 0 hours)? 40
3. Write the equation in the form y = mx + b y = 8x + 40
4. Calculate the predicted score for x = 5 hours: y = 8(5) + 40 80
10.
A regression line has a negative slope. What does this mean?
1. a)As x increases, y decreases
2. b)The line passes through the origin
3. c)As x increases, y increases
4. d)The data has no relationship
Answer: As x increases, y decreases
A negative slope means the line goes downward from left to right. As increases, decreases. This indicates an inverse relationship between the variables.
11.
Find the equation of the line through points and .
Answer: y = 3x + 5
1. Calculate the slope: . What is ? 3
2. The first point is . What is the y-intercept ? 5
3. Write the final equation. y = 3x + 5
12.
A car's fuel efficiency is modeled by , where is the car's weight in kg and is km/L. What is the predicted fuel efficiency for a car weighing 1000 kg?
Answer: 10
km/L. The negative slope shows that heavier cars have lower fuel efficiency.
13.
Find the equation of the line through points and .
Answer: y = -3x + 17
1. Calculate the slope: . What is ? -3
2. Use point and in . Solve: . What is ? 17
3. Write the final equation. y = -3x + 17
14.
Why is it problematic to use a regression equation to predict values far outside the original data range?
1. a)The slope changes automatically
2. b)The equation becomes mathematically invalid
3. c)The y-intercept shifts
4. d)The relationship may not continue linearly beyond the data
Answer: The relationship may not continue linearly beyond the data
This is called extrapolation. While the math works, the real-world relationship may not stay linear outside the data range. For example, fuel efficiency cannot become negative, even if the equation predicts it for very heavy cars.
15.
A house pricing model shows , where is square meters and is price in euros. Find the predicted price for a 120 m² house, and interpret what the slope means.
Answer: 350000
1. What is the slope of the equation? 2500
2. What does the slope represent? (Price per what?) square meter
3. Calculate: . What is the first part, ? 300000
4. Add the base price: . What is the total predicted price? 350000
16.
Two students measured height and shoe size. Student A found a slope of 0.5, and Student B found a slope of 2.0. Both measured the same people. What could explain this difference?
1. a)The data changed between measurements
2. b)They measured at different times
3. c)One made calculation errors
4. d)They used different units for measurement
Answer: They used different units for measurement
The slope depends on the units used. If Student A measured height in meters and Student B in centimeters, the slopes would differ by a factor of 100. This is why it is important to always specify units when reporting regression results.