What did housing prices look like in the "good old days"?...

What did housing prices look like in the "good old days"? The median sale prices for new single family houses are given in the accompanying table for the years 1972 through $1979 .^{\star}$ Letting $Y$ denote the median sales price and $x$ the year (using integers $1,2, \ldots, 8$ ), fit the model $Y=\beta_{0}+\beta_{1} x+\varepsilon .$ What can you conclude from the results?

Key Concepts

Simple Linear Regression

Simple linear regression is a statistical method used to model the relationship between a dependent variable and a single independent variable by fitting a linear equation to observed data. It provides a way to quantify the association between the two variables and offers a straightforward interpretation of how changes in the independent variable are associated with changes in the dependent variable.

Least Squares Estimation

Least squares estimation is the most common method for fitting a regression line in simple linear regression. This technique minimizes the sum of the squared differences between the observed values and the values predicted by the model, resulting in estimates of the regression coefficients that best represent the data.

Interpretation of Regression Coefficients

The regression model includes an intercept (?0) and a slope (?1). The intercept represents the estimated value of the dependent variable when the independent variable is zero, while the slope indicates the expected change in the dependent variable for each one-unit increase in the independent variable. Understanding these coefficients is key in interpreting the underlying relationship between the variables.

Hypothesis Testing in Regression

Hypothesis testing in the context of regression involves assessing whether the estimated slope coefficient is statistically different from zero. A significant slope suggests that there is a meaningful linear relationship between the independent and dependent variables. This is typically done using a t-test to determine if the observed relationship is unlikely to have occurred by random chance.

Trend Analysis

Trend analysis uses regression to determine whether there is an upward or downward trend over time or another ordered variable. By fitting a linear model to the data, one can evaluate if the dependent variable increases or decreases systematically with the independent variable, which is central to understanding long-term patterns or changes within the studied context.

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