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) Plot #1 is just a scatterplot of the response variable (JANTEMP) against the p...

Mar 22, 2024

) Plot #1 is just a scatterplot of the response variable (JANTEMP) against the predictor variable (LONG). (b) Plot #2 gives a scatterplot of the residual values (the difference between the observed and fitted values) against the fitted values. (c) Plot #3 is a scatterplot of the residuals against the predictor variable (the scaled longitude) (d) Plot #4 is a Normal scores plot of the 56 ordered residuals against the corre- sponding Normal scores. (e) all of above are correct

Solution by Steps

step 1

To address the student's question, we need to evaluate the descriptions of the plots provided and determine which, if any, are correct

step 2

Plot #1 is described as a scatterplot of the response variable (JANTEMP) against the predictor variable (LONG). This is a standard way to visualize the relationship between a response variable and a predictor variable in regression analysis

step 3

Plot #2 shows a scatterplot of the residual values against the fitted values. Residuals are the differences between observed values and values predicted by a regression model. Plotting residuals against fitted values is a common diagnostic tool to check for non-linearity, unequal error variances, and outliers

step 4

Plot #3 is a scatterplot of the residuals against the predictor variable (the scaled longitude). This is another common diagnostic plot used to detect non-constant variance (heteroscedasticity) or to identify patterns that may suggest a relationship between the residuals and the predictor variable has not been fully captured by the model

step 5

Plot #4 is a Normal scores plot (or Q-Q plot) of the ordered residuals against the corresponding Normal scores. This plot is used to assess whether the residuals follow a Normal distribution, which is an assumption of many regression models

step 6

Based on the descriptions, all plots are correctly described and represent common practices in regression analysis. Therefore, the correct answer is (e) all of above are correct

Answer

(e) all of above are correct

Key Concept

Diagnostic plots in regression analysis

Explanation

The student's descriptions of the plots correspond to standard diagnostic tools used in regression analysis to check assumptions and evaluate the fit of a model.

\#3 Fitted value \#4

Solution by Steps

step 1

To understand what a fitted value is, we need to know that in linear regression, the fitted value is the estimate of the response variable that the model predicts

step 2

The fitted value

\hat{y}_i

for each observation

i

is calculated using the regression equation

\hat{y}_i = \beta_0 + \beta_1 x_i

, where

\beta_0

is the intercept,

\beta_1

is the slope coefficient, and

x_i

is the predictor variable

step 3

For Plot #4, a Normal scores plot (or Q-Q plot) is used to assess the normality of the residuals. If the residuals are normally distributed, the points should approximately follow a straight line

step 4

The Normal scores are the expected values of the residuals if they were normally distributed. These are plotted against the ordered residuals from the regression

Answer

The fitted value is the predicted value from the regression model, and a Normal scores plot is used to assess the normality of residuals.

Key Concept

Fitted values in regression and Normal scores plot for residuals

Explanation

Fitted values are the predictions made by the regression model, and the Normal scores plot is a graphical tool to check if residuals