Update _25-sa-inf-model-mlr.qmd

exercises/_25-sa-inf-model-mlr.qmd

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-1.  \(a\) (-0.044, 0.346). We are 95% confident that student who go out more than two nights a week on average have GPAs 0.044 points lower to 0.346 points higher than those who do not go out more than two nights a week, when controlling for the other variables in the model. (b) Yes, since the p-value is larger than 0.05 in all cases (not including the intercept).
+1.  \(a\) (-0.044, 0.346). We are 95% confident that student who goes out more than two nights a week on average have GPAs 0.044 points lower to 0.346 points higher than those who do not go out more than two nights a week, when controlling for the other variables in the model. (b) Yes, since the p-value is larger than 0.05 in all cases (not including the intercept).
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 1.  \(a\) `volume` and `diam`; `volume` and `height`; `diam` and `height`. \(b) Each is discernible in its own model. (c) When both `diam`eter and `height` are used in the multiple linear regression model, both continue to be discernible predictors of `volume`.
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-1.  \(a\) **L**inearity: Horror movies seem to show a much different pattern than the other genres. While the residuals plots show a random scatter over years and in order of data collection, there is a clear pattern in residuals for various genres, which signals that this regression model is not appropriate for these data. **I**ndependent observations: The variability of the residuals is higher for data that comes later in the dataset. We don't know if the data are sorted by year, but if so, there may be a temporal pattern in the data that voilates the independence condition. **N**ormality: The residuals are right skewed (skewed to the high end). Constant or **E**qual variability: The residuals vs. predicted values plot reveals some outliers. This plot for only babies with predicted birth weights between 6 and 8.5 pounds looks a lot better, suggesting that for bulk of the data the constant variance condition is met.
+1.  \(a\) **L**inearity: Horror movies seem to show a much different pattern than the other genres. While the residuals plots show a random scatter over years and in order of data collection, there is a clear pattern in residuals for various genres, which signals that this regression model is not appropriate for these data. **I**ndependent observations: The variability of the residuals is higher for data that comes later in the dataset. We don't know if the data are sorted by year, but if so, there may be a temporal pattern in the data that violates the independence condition. **N**ormality: The residuals are right skewed (skewed to the high end). Constant or **E**qual variability: The residuals vs. predicted values plot reveals some outliers. This plot for only babies with predicted birth weights between 6 and 8.5 pounds looks a lot better, suggesting that for bulk of the data the constant variance condition is met.
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 1.  \(a\) Linearity: With so many observations in the dataset, we look for particularly extreme outliers in the histogram of residuals and do not see any. We also don't see a non-linear pattern emerging in the residuals vs. predicted plot. Independent observations: The sample is random and there does not seem to be a trend in the residuals vs. order of data collection plot. Normality: The histogram of residuals appears to be unimodal and symmetic, centered at 0. Constant or equal variability: The residuals vs. predicted values plot reveals some outliers. This plot for only babies with predicted birth weights between 6 and 8.5 pounds looks a lot better, suggesting that for bulk of the data the constant variance condition is met. All concerns raised here are relatively mild. There are some outliers, but there is so much data that the influence of such observations will be minor. (b) $H_0$: The true slope coefficient of habit is zero ($\beta_5 = 0$).  $H_A$: The true slope coefficient of height is different than zero ($\beta_5 \neq 0$). The p-value for the two-sided alternative hypothesis ($\beta_5 \ne 0$) is incredibly 0.0007 (smaller than 0.05), so we reject $H_0$. The data provide convincing evidence that height and weight are positively correlated, given the other variables in the model. The true slope parameter is indeed greater than 0.