Age Gap at Marriage
Is there a relation between the age difference between husband and wife and the percent of a country that is literate? Researchers found the least-squares regression between age difference (husband age minus wife age), y, and literacy rate (percent of the population that is literate), x, is
ŷ = -0.0527x + 7.1
The model applied for 18 ≤ x ≤ 100.
Source: Xu Zhang and Solomon W. Polachek, State University of New York at Binghamton “The Husband-Wife Age Gap at First Marriage: A Cross-Country Analysis.”
a. Interpret the slope.

Verified step by step guidance

1. Identify the variables in the regression equation: \(y\) represents the age difference between husband and wife, and \(x\) represents the literacy rate (percent of the population that is literate).
2. Recall that the slope in a linear regression equation \(\hat{y} = b x + a\) represents the change in the predicted value of \(y\) for a one-unit increase in \(x\).
3. Look at the slope value in the equation \(\hat{y} = -0.0527x + 7.1\), which is \(-0.0527\).
4. Interpret the slope as: For each 1% increase in the literacy rate, the predicted age difference between husband and wife decreases by 0.0527 years.
5. Explain the meaning in context: This suggests that countries with higher literacy rates tend to have smaller age gaps at marriage, according to the model.

Key Concepts

Least-Squares Regression Line

A least-squares regression line models the relationship between two variables by minimizing the sum of the squared differences between observed and predicted values. It is expressed as ŷ = b₀ + b₁x, where ŷ is the predicted response, x is the explanatory variable, b₁ is the slope, and b₀ is the intercept.

Slope Interpretation in Regression

The slope in a regression line represents the average change in the response variable (y) for each one-unit increase in the explanatory variable (x). A negative slope indicates a decrease in y as x increases, while a positive slope indicates an increase.

Contextual Understanding of Variables

Interpreting regression results requires understanding the real-world meaning of variables. Here, x is literacy rate (%) and y is the age difference between husband and wife. This context helps explain what a change in literacy rate implies for the age gap at marriage.