Update _07-sa-model-slr.qmd

exercises/_07-sa-model-slr.qmd

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 1.  \(a\) First calculate the slope: $b_1 = R\times s_y/s_x = 0.636 \times 113 / 99 = 0.726$. Next, make use of the fact that the regression line passes through the point $(\bar{x},\bar{y})$: $\bar{y} = b_0 + b_1 \times \bar{x}$. Plug in $\bar{x}$, $\bar{y}$, and $b_1$, and solve for $b_0$: 51. Solution: $\widehat{travel~time} = 51 + 0.726 \times distance$. (b) $b_1$: For each additional mile in distance, the model predicts an additional 0.726 minutes in travel time. $b_0$: When the distance travelled is 0 miles, the travel time is expected to be 51 minutes. It does not make sense to have a travel distance of 0 miles in this context. Here, the $y$-intercept serves only to adjust the height of the line and is meaningless by itself. (c) $R^2 = 0.636^2 = 0.40$. About 40% of the variability in travel time is accounted for by the model, i.e., explained by the distance travelled. (d) $\widehat{travel~time} = 51 + 0.726 \times distance  = 51 + 0.726 \times 103 \approx 126$ minutes. (Note: we should be cautious in our predictions with this model since we have not yet evaluated whether it is a well-fit model.) (e) $e_i = y_i - \hat{y}_i = 168 - 126 = 42$ minutes. A positive residual means that the model underestimates the travel time. (f) No, this calculation would require extrapolation.
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-1.  \(a\) $\widehat{\texttt{poverty}} = 4.60 + 2.05 \times \texttt{unemployment\_rate}.$ (b) The model predicts a poverty rate of 4.60\% for counties with 0\% unemployment, on average. This is not a meaningful value as no counties have such low unexmployment, it just serves to adjust the height of the regression line. (c) For each additional percentage increase in unemployment rate, poverty rate is predicted to be higher, on average, by 2.05\%. (d) Unemployment rate explains 46\% of the variability in poverty levels in US counties. (e) $\sqrt{0.46} = 0.678.$
+1.  \(a\) $\widehat{\texttt{poverty}} = 4.60 + 2.05 \times \texttt{unemployment\_rate}.$ (b) The model predicts a poverty rate of 4.60\% for counties with 0\% unemployment, on average. This is not a meaningful value as no counties have such low unemployment, it just serves to adjust the height of the regression line. (c) For each additional percentage increase in unemployment rate, poverty rate is predicted to be higher, on average, by 2.05\%. (d) Unemployment rate explains 46\% of the variability in poverty levels in US counties. (e) $\sqrt{0.46} = 0.678.$
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 1.  \(a\) There is an outlier in the bottom right. Since it is far from the center of the data, it is a point with high leverage. It is also an influential point since, without that observation, the regression line would have a very different slope. (b) There is an outlier in the bottom right. Since it is far from the center of the data, it is a point with high leverage. However, it does not appear to be affecting the line much, so it is not an influential point. (c) The observation is in the center of the data (in the x-axis direction), so this point does *not* have high leverage. This means the point won't have much effect on the slope of the line and so is not an influential point.