AP Stats Regression Analysis 3 — Questions and Answers
Question 1: In the regression model ŷ = b₀ + b₁x, what does b₁ represent?
- The y-intercept
- The predicted value of y when x = 0
- The average change in y for each one-unit increase in x (Correct answer)
- The correlation coefficient
Correct answer: The average change in y for each one-unit increase in x
b₁ is the slope, representing the estimated average change in the response variable y for each one-unit increase in the explanatory variable x.
Question 2: A researcher finds that the correlation between shoe size and reading ability in children is r = 0.78. The best explanation is:
- Larger feet cause better reading ability
- Better reading ability causes foot growth
- Age is a lurking variable that increases both shoe size and reading ability (Correct answer)
- The correlation must be incorrect because it is too high
Correct answer: Age is a lurking variable that increases both shoe size and reading ability
Age is a confounding (lurking) variable — older children have both larger feet and greater reading ability, creating a spurious correlation.
Question 3: In a regression context, an influential observation is best described as a point that:
- Has a large residual only
- Has high leverage and notably changes the regression line when removed (Correct answer)
- Falls exactly on the regression line
- Has x and y both equal to their means
Correct answer: Has high leverage and notably changes the regression line when removed
An influential observation has high leverage (extreme x-value) and substantially affects the slope and/or intercept when included or excluded.
Question 4: If all points in a scatterplot lie exactly on a straight line with negative slope, what is the value of r?
- 0
- 1
- −1 (Correct answer)
- 0.5
Correct answer: −1
A perfect negative linear relationship gives r = −1.
Question 5: The standard error of the slope (SE_b₁) in regression is used to:
- Calculate r²
- Construct a confidence interval or conduct a t-test for the slope (Correct answer)
- Find the y-intercept
- Measure the total variation in y
Correct answer: Construct a confidence interval or conduct a t-test for the slope
SE_b₁ is used to form t-intervals and t-tests to make inferences about the true population slope β₁.
Question 6: A scatterplot shows a fan-shaped (funnel) pattern in the residuals. Which condition for regression inference is violated?
- Linearity
- Independence
- Normality of residuals
- Equal variance (homoscedasticity) (Correct answer)
Correct answer: Equal variance (homoscedasticity)
A fan-shaped residual pattern indicates non-constant variance (heteroscedasticity), violating the equal-variance condition.
Question 7: The least-squares regression line always passes through which point?
- The origin (0, 0)
- The point (x̄, ȳ) (Correct answer)
- The point with the largest x-value
- The point with the smallest residual
Correct answer: The point (x̄, ȳ)
The least-squares line always passes through (x̄, ȳ), the point of means.
AP Stats Regression Analysis 3
In the regression model ŷ = b₀ + b₁x, what does b₁ represent?