![]() ![]() Therefore, $r^2$ for this data set is much smaller than $r^2$ for the data set in (a).įigure 8.12 - The data in (a) results in a high value of $r^2$, while the data shown in (b) results in a low value of $r^2$.įor the data in Example 8.31, find the coefficient of determination. On the other hand, for the data shown in (b), a lot of variation in $y$ is left unexplained by the regression model. \textrm$'s are relatively close to the $y_i$'s, so $r^2$ is close to $1$. The other variable, denoted y, is regarded as the response, outcome, or dependent variable. ![]() First, we take expectation from both sides to obtain Simple linear regression is a statistical method that allows us to summarize and study relationships between two continuous (quantitative) variables: One variable, denoted x, is regarded as the predictor, explanatory, or independent variable. Where $\epsilon$ is a $N(0,\sigma^2)$ random variable independent of $X$. ![]() Here, we assume that $x_i$'s are observed values of a random variable $X$. For simple linear regression, the least squares estimates of the model parameters 0 and 1 are denoted b0 and b1. ![]()
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