chp 24 - added and fixed exercises

24-inf-model-slr.qmd

@@ -947,7 +947,7 @@ When fitting a least squares line, we generally require the following:
 -   **Nearly normal residuals.** Generally, the residuals should be nearly normal.
     When this condition is found to be unreasonable, it is often because of outliers or concerns about influential points, which we'll talk about more in @sec-outliers-in-regression.
     An example of a residual that would be potentially concerning is shown in the second panel of @fig-whatCanGoWrongWithLinearModel, where one observation is clearly much further from the regression line than the others.  Outliers should be treated extremely carefully.  Do not automatically remove an outlier if it truly belongs in the dataset.  However, be honest about its impact on the analysis.  A strategy for dealing with outliers is to present two analyses: one with the outlier and one without the outlier.
-    Additionally, a type of violation of normality happens when the positive residuals are smaller in magnitude than the negative residuals (or vice versa).  That is, when the residuals are not symmetrically distributed around the line $y=0.$ 
+Additionally, a type of violation of normality happens when the positive residuals are smaller in magnitude than the negative residuals (or vice versa).  That is, when the residuals are not symmetrically distributed around the line $y=0.$ 
 
 -   **Constant or equal variability.** The variability of points around the least squares line remains roughly constant.
     An example of non-constant variability is shown in the third panel of @fig-whatCanGoWrongWithLinearModel, which represents the most common pattern observed when this condition fails: the variability of $y$ is larger when $x$ is larger.
@@ -1180,8 +1180,6 @@ If there are large deviations, we will be unable to trust the calculated p-value
 The linearity condition is among the most important if your goal is to understand a linear model between $x$ and $y$.
 For example, the value of the slope will not be at all meaningful if the true relationship between $x$ and $y$ is quadratic, as in @fig-notGoodAtAllForALinearModel.
 Not only should we be cautious about the inference, but the model *itself* is also not an accurate portrayal of the relationship between the variables.
-
-In @sec-inf-model-mlr we discuss model modifications that can often lead to an excellent fit of strong relationships other than linear ones.
 However, an extended discussion on the different methods for modeling functional forms other than linear is outside the scope of this text.
 
 **The importance of Independence**
@@ -1207,7 +1205,7 @@ You should consider the "bell" of the normal distribution as sitting on top of t
 The normality condition is less important than linearity or independence for a few reasons.
 First, the linear model fit with least squares will still be an unbiased estimate of the true population model.
 However, the distribution of the estimate will be unknown.
-Fortunately the Central Limit Theorem tells us that most of the analyses (e.g., SEs, p-values, confidence intervals) done using the mathematical model (with the $t$-distribution) will still hold (even if the data are not normally distributed around the line) as long as the sample size is large enough.
+Fortunately the Central Limit Theorem (described in @sec-one-mean-math) tells us that most of the analyses (e.g., SEs, p-values, confidence intervals) done using the mathematical model (with the $t$-distribution) will still hold (even if the data are not normally distributed around the line) as long as the sample size is large enough.
 One analysis method that *does* require normality, regardless of sample size, is creating intervals which predict the response of individual outcomes at a given $x$ value, using the linear model.
 One additional reason to worry slightly less about normality is that neither the randomization test nor the bootstrapping procedures require the data to be normal around the line.
 
@@ -1229,7 +1227,7 @@ In particular, random effects models, repeated measures, and interaction are all
 When the technical conditions hold, the extensions to the linear model can provide important insight into the data and research question at hand.
 We will discuss some of the extended modeling and associated inference in @sec-inf-model-mlr and @sec-inf-model-logistic.
 Many of the techniques used to deal with technical condition violations are outside the scope of this text, but they are taught in universities in the very next class after this one.
-If you are working with linear models or curious to learn more, we recommend that you continue learning about statistical methods applicable to a larger class of datasets.
+If you are working with linear models or are curious to learn more, we recommend that you continue learning about statistical methods applicable to a larger class of datasets.
 
 \clearpage