Explaining Slopes and Percentage Changes

Imagine you're a data analyst working for a company that sells high-end coffee machines. The company wants to understand how the amount of coffee sold per month (in units) is related to the amount spent on advertising (in thousands of dollars) each month. You decide to use a linear model to analyze this relationship. You collect data over several months and find that the monthly units sold can be predicted fairly accurately using the following linear model: Units Sold=50+8×(Advertising Expenditure) where: The intercept (constant term) is the number of units sold when the advertising expenditure is $0. The slope (coefficient of the Advertising Expenditure) represents the increase in units sold for every additional thousand dollars spent on advertising. What does the intercept of the linear model represent in the context of this scenario?

• The amount of advertising expenditure needed to achieve 50 coffee machines sold.
• The increase in the number of coffee machines sold for each additional $1,000 spent on advertising.
• The total number of coffee machines sold were 50 when the advertising expenditure is $0.
• The advertising expenditure that results in selling 50 coffee machines.

If a slope of a linear model is 2, what does it imply about the relationship between the variables?

• A) For every unit increase in the independent variable, the dependent variable increases by 2 units.
• B) For every 2 units increase in the independent variable, the dependent variable decreases by 1 unit.
• C) The independent variable stays constant while the dependent variable decreases.
• D) The relationship between the variables is random and not linear.

In a linear regression model, how would you interpret a slope of -3?

• A) For every 1 unit increase in the independent variable, the dependent variable decreases by 3 units.
• B) For every 3 unit increase in the independent variable, the dependent variable increases by 1 unit.
• C) The dependent variable increases by 3 units for every 1 unit increase in the independent variable.
• D) The independent variable stays constant while the dependent variable increases.

Which of the following statements is true regarding a residual plot in linear regression analysis?

• A residual plot should always show a perfectly straight line with no deviations.
• A residual plot helps us assess the randomness and of residuals.
• In a residual plot, the horizontal axis represents the independent variable.
• Residual plots are only used when the correlation coefficient is 1.

A taxi company charges a flat fee of $5 plus $2 per mile driven. Write the linear equation for this scenario and explain what the slope represents.

• y=2x+5. The slope of 2 represents the flat fee charged by the taxi company.
• y=2x+5. The slope of 2 represents the cost per mile driven.
• y=5x+2. The slope of 5 represents the cost per mile driven.
• y=5x+2. The slope of 5 represents the flat fee charged by the taxi company.

A gym membership costs $40 to sign up and then $20 per month. Write the equation for the total cost of the membership over time and explain the meaning of the slope.

• y=20x+40. The slope of 20 represents the total cost at the start.
• y=40x+20. The slope of 40 represents the monthly membership cost.
• y=20x+40. The slope of 20 represents the monthly membership cost.
• y=40x+20. The slope of 40 represents the total cost after one month.

A car rental company charges a base fee of $30 and $10 per hour for renting a car. How would you explain the meaning of the slope in this linear equation: y=10x+30?

• A) The slope of 10 means the total cost decreases by $10 for every hour.
• B) The slope of 10 means the total cost increases by $10 for every hour.
• C) The slope of 30 means the hourly rate is $30.
• D) The slope of 30 means the cost increases by $30 every hour.

The temperature of a cup of coffee decreases by 5°F every minute after being left on a table. Write the linear equation and explain what the slope represents in this situation.

• y=−5x+100. The slope of -5 represents the initial temperature of the coffee.
• y=5x+100. The slope of 5 represents the rate of cooling per minute.
• y=−5x+100. The slope of -5 represents the decrease in temperature per minute.
• y=100x−5. The slope of -5 represents the final temperature of the coffee.

A landscaper charges $100 for equipment rental and $15 for every hour of work. Write the equation for the total cost and explain what the slope represents.

• y=15x+100. The slope of 15 represents the cost for each hour of labor.
• y=100x+15. The slope of 100 represents the cost for each hour of labor.
• y=15x+100. The slope of 15 represents the total cost for the equipment.
• y=100x+15. The slope of 100 represents the number of hours worked.

A population of bacteria grows according to the exponential model P(t)=100⋅(1.08)^t, where P(t) represents the population at time t in hours. What does the 1.08 in the equation represent?

• A) The population decreases by 8% each hour.
• B) The population increases by 8% each hour.
• C) The population increases by 108 bacteria each hour.
• D) The population decreases by 108 bacteria each hour.