Analyzing Residuals and Graphing Linear Relationships

A local gym tracks the number of members over the years. The data shows a linear relationship between the year and the number of members. If the regression equation is $y = 50x + 200$, where $x$ is the number of years since 2010, how many members are expected in 2023?

• 750

• 1000

• 900

• 850

The same gym wants to analyze how well their regression model predicts actual membership numbers. In 2022, they had 350 members. Calculate the residual for that year using the regression equation from the previous question.

• -50

• -200

• 50

• 100

A car rental company finds that the cost of renting a car can be modeled by the equation $y = 30x + 50$, where $y$ is the total cost and $x$ is the number of days rented. Graph this linear relationship and identify the y-intercept and slope.

• Slope: 30, Y-intercept: 40

• Slope: 20, Y-intercept: 50

• Slope: 50, Y-intercept: 30

• Slope: 30, Y-intercept: 50

A student collects data on the number of hours studied and test scores. The regression line is given by the equation $y = 5x + 60$. If a student studies for 4 hours, what is the predicted test score?

• 80

• 100

• 70

• 90

Using the same regression line from the previous question, if the actual test score for the student who studied for 4 hours was 78, what is the residual?

• -4

• -6

• 2

• 10

A researcher is studying the relationship between temperature and ice cream sales. The regression equation is $y = 10x + 200$, where $x$ is the temperature in degrees Celsius. If the temperature is 25 degrees, how many ice creams are expected to be sold?

• 300

• 500

• 400

• 450

In the ice cream sales problem, if the actual sales on that day were 270, calculate the residual. What does this residual indicate about the prediction?

• -30

• 100

• -50

• 0

A school is analyzing the relationship between hours of sleep and student performance. The regression equation is $y = 3x + 70$. If a student sleeps for 6 hours, what is the predicted performance score?

• 75

• 80

• 92

• 88

If the actual performance score for the student who slept for 6 hours was 75, what is the residual? Discuss what this means in the context of the regression model.

• The residual is -5.

• The residual is -15.

• The residual is 10.

• The residual is 0.

A farmer is studying the relationship between the amount of fertilizer used and crop yield. The regression equation is $y = 2x + 50$. If the farmer uses 10 units of fertilizer, what is the predicted crop yield?

• 80

• 60

• 70

• 90