Question 1

In regression, it is desired to predict the dependent variable based on values of other related independent variables. Occasionally, there are relationships that exist between the independent
Variables. Which of the following multiple regression pitfalls does this example describe?
A)Multicollinearity
B)Extrapolation
C)Estimability
D)Stepwise Regression

Accepted Answer

Multicollinearity refers to the situation in regression analysis where independent variables are highly correlated with each other. This can make it difficult to determine the individual effect of each independent variable on the dependent variable.

Question 2

The model $E ( y ) = \beta _ { 0 } + \beta _ { 1 } ^ { x }$ was fit to a set of data, and the following plot of residuals against $x$ values was obtained.

Interpret the residual plot.

Accepted Answer

The answer of The model \(E ( y ) =...

Question 3

The number of levels of observed x-values must be equal to the order of the polynomial in x that
you want to fit.

Accepted Answer

The order of the polynomial you want to fit is independent of the number of levels of observed x-values. You can fit a polynomial of any order as long as you have more data points than the degree of the polynomial plus one.

Question 4

The printout below shows part of the least squares regression analysis for the model $E ( y ) = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 }$ fit to a set of data. The model attempts to predict a score on the final exam in a statistics course based on the scores on the first two tests in the class.
ANOVA
$\begin{array}{llllll}
\hline & d f & S S & M S & F & \text { Significance F } \
\hline \text { Regression } & 2 & 1293.125328 & 646.5626641 & 21.27366772 & 2.35769 \mathrm{E}-05 \
\text { Residual } & 17 & 516.6746719 & 30.39262776 & & \
\text { Total } & 19 & 1809.8 & & & \
\hline
\end{array}$

$\begin{array}{lllllll}
\hline & \text { Coefficients } & \text { Standard Error } & t \text { Stat } & \text { P-value } & \text { Lower 95\% } & \text { Upper 95\% } \
\hline \text { Intercept } & -4.409686163 & 16.72267106 & -0.263695085 & 0.795184685 & -39.69148734 & 30.87211502 \
\text { Test 1 } & 0.397435806 & 0.343012569 & 1.158662514 & 0.262611745 & -0.326258467 & 1.121130079 \
\text { Test 2 } & 0.638805278 & 0.224623383 & 2.843894834 & 0.011217936 & 0.164890704 & 1.112719852 \
\hline
\end{array}$
 Is there evidence of multicollinearity in the printout? Explain.

Accepted Answer

The answer of The printout below shows part of the...

Question 5

Which of the following is not a possible indicator of multicollinearity?
A) signs opposite from what is expected in the estimated $\beta$ parameters
B) non-significant $t$-tests for individual $\beta$ parameters when the $F$-test for overall model adequacy is significant
C) significant correlations between pairs of independent variables
D) non-random patterns in the plot of the residuals versus the fitted values

Accepted Answer

Non-random patterns in the plot of the residuals versus the fitted values indicate issues with the model fit or heteroscedasticity, not multicollinearity. Multicollinearity concerns the correlation between independent variables, not the relationship between residuals and fitted values.

Question 6

It is dangerous to predict outside the range of the data collected in a regression analysis. For instance, we shouldnʹt predict the price of a 5000 square foot home if all our sample homes were
Smaller than 4500 square feet. Which of the following multiple regression pitfalls does this example
Describe?
A)Multicollinearity
B)Extrapolation
C)Estimability
D)Stepwise Regression

Accepted Answer

Extrapolation refers to making predictions outside the range of the data. In this case, predicting the price of a home larger than any in the sample (5000 square feet when all sample homes are smaller than 4500 square feet) is an example of extrapolation, which can lead to inaccurate or unreliable predictions.

Question 7

The concessions manager at a beachside park recorded the high temperature, the number
of people at the park, and the number of bottles of water sold for each of 12 consecutive
Saturdays. The data are shown below. $\begin{array}{ccc}
\hline \text { Bottles Sold } & \text { Temperature }\left({ }^{\circ} \mathrm{F}\right) & \text { People } \
\hline 341 & 73 & 1625 \
425 & 79 & 2100 \
457 & 80 & 2125 \
485 & 80 & 2800 \
469 & 81 & 2550 \
395 & 82 & 1975 \
511 & 83 & 2675 \
549 & 83 & 2800 \
543 & 85 & 2850 \
537 & 88 & 2775 \
621 & 89 & 2800 \
897 & 91 & 3100 \
\hline
\end{array}$

a. Fit the model $E ( y ) = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \beta _ { 3 } x _ { 1 } x _ { 2 }$ to the data, letting $y$ represent the number of bottles of water sold, $x _ { 1 }$ the temperature, and $x _ { 2 }$ the number of people at the park.
b. Identify at least two indicators of multicollinearity in the model.
c. Comment on the usefulness of the model to predict the number of bottles of water sold on a Saturday when the high temperature is $103 ^ { \circ } \mathrm { F }$ and there are 3500 people at the park.

Accepted Answer

The answer of The concessions manager at a beachside park...

Question 8

In the presence of multicollinearity, the predicted values of $\hat { y }$ are actually quite good for values of $x$ far outside the range of the sampled values of $x$.

Accepted Answer

Multicollinearity primarily affects the precision of the estimated coefficients, not the fit of the model or predictions within the range of observed values. Predictions outside the observed range of $x$ (extrapolation) are generally unreliable, regardless of multicollinearity.

Question 9

In the presence of multicollinearity, you should avoid making inferences about the parameters
based on the t-tests.

Accepted Answer

Multicollinearity among predictors in a regression model can inflate the variance of the coefficient estimates, leading to unreliable t-tests and making it difficult to assess the individual effect of each predictor accurately.

Question 10

The model $E ( y ) = \beta _ { 0 } + \beta _ { 1 } x$ was fit to a set of data, and the following plot of residuals against $x$ values was obtained.

Accepted Answer

The answer of The model \(E ( y ) =...

Question 11

The model $E ( y ) = \beta _ { 0 } + \beta _ { 1 } x$ was fit to a set of data, and the following plot of residuals against $x$ values was obtained.

Interpret the residual plot.

Accepted Answer

The answer of The model \(E ( y ) =...

Quiz 12: Multiple Regression and Model Building

In regression, it is desired to predict the dependent variable based on values of other related independent variables. Occasionally, there are relationships that exist between the independent Variables. Which of the following multiple regression pitfalls does this example describe?

The number of levels of observed x-values must be equal to the order of the polynomial in x that you want to fit.

Which of the following is not a possible indicator of multicollinearity?

In the presence of multicollinearity, the predicted values of $\hat { y }$ are actually quite good for values of $x$ far outside the range of the sampled values of $x$ .

In the presence of multicollinearity, you should avoid making inferences about the parameters based on the t-tests.

The model $E ( y ) = \beta _ { 0 } + \beta _ { 1 } x$ was fit to a set of data, and the following plot of residuals against $x$ values was obtained. $The model E ( y ) = \beta _ { 0 } + \beta _ { 1 } x was fit to a set of data, and the following plot of residuals against x values was obtained.$

Quiz 12: Multiple Regression and Model Building

In regression, it is desired to predict the dependent variable based on values of other related independent variables. Occasionally, there are relationships that exist between the independent Variables. Which of the following multiple regression pitfalls does this example describe?

The model E(y)=β0+β1xE ( y ) = \beta _ { 0 } + \beta _ { 1 } ^ { x }E(y)=β0​+β1x​ was fit to a set of data, and the following plot of residuals against xxx values was obtained. Interpret the residual plot.

The number of levels of observed x-values must be equal to the order of the polynomial in x that you want to fit.

Which of the following is not a possible indicator of multicollinearity?

In the presence of multicollinearity, the predicted values of y^\hat { y }y^​ are actually quite good for values of xxx far outside the range of the sampled values of xxx .

In the presence of multicollinearity, you should avoid making inferences about the parameters based on the t-tests.

The model E(y)=β0+β1xE ( y ) = \beta _ { 0 } + \beta _ { 1 } xE(y)=β0​+β1​x was fit to a set of data, and the following plot of residuals against xxx values was obtained.

The model E(y)=β0+β1xE ( y ) = \beta _ { 0 } + \beta _ { 1 } xE(y)=β0​+β1​x was fit to a set of data, and the following plot of residuals against xxx values was obtained. Interpret the residual plot.

In the presence of multicollinearity, the predicted values of $\hat { y }$ are actually quite good for values of $x$ far outside the range of the sampled values of $x$ .

The model $E ( y ) = \beta _ { 0 } + \beta _ { 1 } x$ was fit to a set of data, and the following plot of residuals against $x$ values was obtained. $The model E ( y ) = \beta _ { 0 } + \beta _ { 1 } x was fit to a set of data, and the following plot of residuals against x values was obtained.$