Analyzing a Newspaper ArticleA Chicago Tribune article claimed that school districts with higher poverty levels have shorter school days.
b. The least-squares regression line for length, y, and income, x, is y^=−0.0102x+7.11. Explain the meaning of the slope for this regression line. Should the y-intercept be interpreted? If it makes sense, interpret it.

Verified step by step guidance

1. Identify the variables involved: here, \( y \) represents the length of the school day, and \( x \) represents income (likely a measure related to poverty level).
2. Recall that the slope in a regression line \( \hat{y} = b_1 x + b_0 \) represents the estimated change in \( y \) for a one-unit increase in \( x \). In this case, the slope is \( -0.0102 \), which means for each one-unit increase in income, the length of the school day changes by \( -0.0102 \) units.
3. Interpret the slope in context: since the slope is negative, it suggests that as income increases, the length of the school day decreases. This aligns with the claim that higher poverty (lower income) is associated with longer school days.
4. Consider the y-intercept \( b_0 = 7.11 \), which represents the predicted length of the school day when income \( x = 0 \). Evaluate whether \( x = 0 \) is a meaningful or realistic value in this context. If \( x = 0 \) is outside the range of observed data or not meaningful, the y-intercept should not be interpreted practically.
5. If the y-intercept is interpretable, explain it as the estimated length of the school day for a school district with zero income. Otherwise, state that it is a mathematical artifact of the regression line without practical interpretation.

Key Concepts

Least-Squares Regression Line

The least-squares regression line is a straight line that best fits the data by minimizing the sum of the squared differences between observed and predicted values. It models the relationship between an independent variable (x) and a dependent variable (y), allowing predictions of y based on x.

Interpretation of the Slope

The slope of a regression line represents the average change in the dependent variable (y) for each one-unit increase in the independent variable (x). A negative slope indicates that as x increases, y tends to decrease, quantifying the strength and direction of the relationship.

Interpretation of the Y-Intercept

The y-intercept is the predicted value of y when x equals zero. Its interpretation depends on whether x=0 is meaningful in context; if not, the intercept may lack practical significance and should be interpreted cautiously or not at all.