Income and Education
In Problem 15 from Section 4.1, a scatter diagram and correlation coefficient suggested there is a linear relation between the percentage of individuals who have at least a bachelor's degree and median income in the states. In fact, the least-squares regression equation is ŷ = 1103x + 31,955 where y is the median income and x is the percentage of individuals 25 years and older with at least a bachelor's degree in the state.
c. Interpret the slope.

Verified step by step guidance

1. Identify the variables involved: here, \(x\) represents the percentage of individuals aged 25 and older with at least a bachelor's degree, and \(y\) represents the median income in the state.
2. Recall that the slope in a least-squares regression equation \(\hat{y} = b x + a\) represents the estimated change in the response variable \(y\) for each one-unit increase in the explanatory variable \(x\).
3. Look at the given regression equation: \(\hat{y} = 1103x + 31,955\). The slope is 1103, which means for every 1% increase in the percentage of individuals with at least a bachelor's degree, the median income changes by 1103 dollars.
4. Interpret the slope in context: this means that as the percentage of college-educated individuals increases by one percentage point, the median income in the state is expected to increase by approximately \$1103, assuming the linear model is appropriate.
5. Note that this interpretation assumes all other factors remain constant and that the relationship is linear as suggested by the regression model.

Key Concepts

Least-Squares Regression Equation

The least-squares regression equation models the linear relationship between two variables by minimizing the sum of squared differences between observed and predicted values. It is expressed as ŷ = b₁x + b₀, where b₁ is the slope and b₀ is the y-intercept, allowing prediction of the dependent variable based on the independent variable.

Slope Interpretation in Regression

The slope in a regression equation represents the average change in the dependent variable for each one-unit increase in the independent variable. In this context, it quantifies how much median income is expected to increase when the percentage of individuals with at least a bachelor's degree rises by one percentage point.

Contextual Understanding of Variables

Interpreting regression results requires understanding the real-world meaning of variables involved. Here, x is the percentage of adults with a bachelor's degree, and y is median income, so the slope's interpretation connects educational attainment to income levels, providing insight into socioeconomic trends.