Least-Squares Regression: Interpreting and Applying the Regression Line

Q1. Predict the median income of a state in which 25% of adults 25 years and older have at least a bachelor's degree.

Background

Topic: Least-Squares Regression Line

This question tests your ability to use a regression equation to make predictions about the dependent variable (median income) based on a given value of the independent variable (percentage with a bachelor's degree).

Key Terms and Formulas

• Regression Equation: $\hat{y} = 1.249x + 18.29$

• $\hat{y}$ = predicted median income (in thousands of dollars)

• $x$ = percentage of adults 25+ with at least a bachelor's degree

Step-by-Step Guidance

1. Identify the value of $x$ to use in the regression equation. Here, $x = 25$.

2. Substitute $x = 25$ into the regression equation: $\hat{y} = 1.249x + 18.29$.

3. Set up the calculation: $\hat{y} = 1.249 \times 25 + 18.29$.

4. Multiply $1.249$ by $25$ to find the first term.

5. Add the result to $18.29$ to get the predicted median income (in thousands of dollars).

Try solving on your own before revealing the answer!

Final Answer:

$\hat{y} = 1.249 \times 25 + 18.29 = 31.225 + 18.29 = 49.515$

The predicted median income is $49,515.

Q2. In Virginia, 41.8% of adults 25 years and older have at least a bachelor's degree. The median income in Virginia is $66,000. Is this income higher or lower than what you would predict? Why?

Background

Topic: Comparing Observed and Predicted Values Using Regression

This question asks you to use the regression equation to predict the median income for a given percentage, then compare it to the actual observed value to determine if it is higher or lower than predicted.

Key Terms and Formulas

• Regression Equation: $\hat{y} = 1.249x + 18.29$

• Observed Value: Actual median income ($y$)

• Predicted Value: $\hat{y}$ from the regression equation

• Residual: $y - \hat{y}$

Step-by-Step Guidance

1. Identify the value of $x$ for Virginia: $x = 41.8$.

2. Substitute $x = 41.8$ into the regression equation: $\hat{y} = 1.249 \times 41.8 + 18.29$.

3. Calculate $1.249 \times 41.8$ to find the first term.

4. Add the result to $18.29$ to get the predicted median income (in thousands of dollars).

5. Compare the predicted value to the actual median income ($66,000$) to determine if it is higher or lower than predicted.

Try solving on your own before revealing the answer!

Final Answer:

$\hat{y} = 1.249 \times 41.8 + 18.29 = 52.202 + 18.29 = 70.492$

The predicted median income is $70,492$, which is higher than the actual median income of $66,000$. Therefore, Virginia's actual income is lower than what the regression model predicts for its education level.

Q3. Interpret the slope.

Background

Topic: Interpreting Regression Coefficients

This question tests your understanding of what the slope of the regression line means in the context of the data.

Key Terms and Formulas

• Slope ($m$): The amount by which the predicted $y$ changes for each one-unit increase in $x$.

• Here, $m = 1.249$.

Step-by-Step Guidance

1. Recall that the slope represents the change in the predicted median income for each additional percentage point of adults with a bachelor's degree.

2. Express this in the context of the problem: For each 1% increase in adults with a bachelor's degree, the predicted median income increases by the value of the slope (in thousands of dollars).

3. State the units: The slope is measured in thousands of dollars per percentage point.

Try solving on your own before revealing the answer!

Final Answer:

The slope of $1.249$ means that for each 1% increase in the percentage of adults 25 years and older with at least a bachelor's degree, the predicted median income increases by $1,249$.

Q4. Explain why it does not make sense to interpret the intercept.

Background

Topic: Interpreting the y-Intercept in Regression

This question asks you to consider the meaning of the y-intercept in the context of the data and whether it is meaningful to interpret it literally.

Key Terms and Formulas

• y-Intercept ($b$): The predicted value of $y$ when $x = 0$.

• Here, $b = 18.29$ (in thousands of dollars).

Step-by-Step Guidance

1. Recall that the y-intercept represents the predicted median income when the percentage of adults with a bachelor's degree is 0%.

2. Consider whether it is realistic or meaningful for a state to have 0% of adults with a bachelor's degree.

3. Think about whether the regression model is valid for values of $x$ far outside the range of the observed data.

Try solving on your own before revealing the answer!

Final Answer:

It does not make sense to interpret the intercept because it predicts the median income for a state with 0% of adults having a bachelor's degree, which is not realistic or observed in the data. The regression model should not be used to extrapolate far outside the range of the data.