Question 1

A regression residual is the difference between an observed y value and its corresponding predicted value.

Accepted Answer

This is the definition of a regression residual.

Question 2

In any production process in which one or more workers are engaged in a variety of tasks, the total time spent in production varies as a function of the size of the workpool and the level of output of the various activities. In a large metropolitan department store, it is believed that the number of man-hours worked $( y )$ per day by the clerical staff depends on the number of pieces of mail processed per day $\left( x _ { 1 } \right)$ and the number of checks cashed per day $\left( x _ { 2 } \right)$. Data collected for $n = 20$ working days were used to fit the model:
$$E ( y ) = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 }$$
A partial printout for the analysis follows:
$\quad $$\quad $$\quad $$\quad $$\quad $$\quad $$\quad $$\quad $$\quad $$\quad $$\quad $$\quad $$\quad $$\quad $$\text{Parameter Estimates}$
$\begin{array} { l r r r r r } & & \text { PARAMETER } & \text { STANDARD } & \text { T FOR 0: } & \ \text { VARIABLE } & \text { DF } & \text { ESTIMATE } & \text { ERROR } & \text { PARAMETER = 0 } & \text { PROB > |T| } \ & & & & & \ \text { INTERCEPT } & 1 & 114.420972 & 18.6848744 & 6.124 & 0.0001 \ \text { X1 } & 1 & - 0.007102 & 0.00171375 & - 4.144 & 0.0007 \ \text { X2 } & 1 & 0.037290 & 0.02043937 & 1.824 & 0.0857 \end{array}$
Calculate a $95 \%$ confidence interval for $\beta _ { 1 }$.
A) $- 4.144 \pm .0007$
B) $- .007 \pm .0007$
C) $- .007 \pm .0017$
D) $- .007 \pm .0036$

Accepted Answer

The answer of In any production process in which one...

Question 3

As part of a study at a large university, data were collected on $n = 224$ freshmen computer science (CS) majors in a particular year. The researchers were interested in modeling $y$, a student's grade point average (GPA) after three semesters, as a function of the following independent variables (recorded at the time the students enrolled in the university):
$x _ { 1 } =$ average high school grade in mathematics (HSM)
$x _ { 2 } =$ average high school grade in science (HSS)
$x _ { 3 } =$ average high school grade in English (HSE)
$x _ { 4 } =$ SAT mathematics score (SATM)
$x _ { 5 } =$ SAT verbal score (SATV)

A first-order model was fit to data.

A $95 \%$ confidence interval for $\beta _ { 1 }$ is $( .06 , .22 )$. Interpret this result.

A) $95 \%$ of the GPAs fall within .06 to $.22$ of their true values.
B) We are $95 \%$ confident that a CS freshman's HS math grade increases by an amount between $.06$ and $.22$ for every 1-point increase in GPA, holding $x _ { 2 } - x _ { 5 }$ constant.
C) We are $95 \%$ confident that the mean GPA of all CS freshmen after three semesters falls between 06 and .22.
D) We are $95 \%$ confident that a CS freshman's GPA increases by an amount between .06 and .22 for every 1-point increase in average HS math grade, holding $x _ { 2 } - x _ { 5 }$ constant.

Accepted Answer

The confidence interval for $\beta_1$ suggests that we are 95% confident that the GPA increases by an amount between 0.06 and 0.22 for every 1-point increase in the average high school math grade, holding other variables constant.

Question 4

The stepwise regression model should not be used as the final model for predicting y.

Accepted Answer

Stepwise regression can be useful for selecting variables to include in a model, but the resulting model may not be the best overall predictor of y. There are other methods, such as ridge regression or LASSO, that can be used to improve the accuracy and reliability of the final predictive model.

Question 5

The independent variables x₁ and x₂ interact when the effect on E(y) of a change in x₁ depends on x₂.

Accepted Answer

Interaction between variables means that the effect of one variable on the dependent variable depends on the level of another variable.

Question 6

The rejection of the null hypothesis in a global F-test means that the model is the best model for providing reliable estimates and predictions.

Accepted Answer

The rejection of the null hypothesis in a global F-test only suggests that there is statistically significant evidence that at least one of the coefficients in the model is different from zero. It does not necessarily mean that the model is the best or only model for providing reliable estimates and predictions. Other factors such as model complexity, goodness of fit, and predictive accuracy should also be considered.

Question 7

Retail price data for $n = 60$ hard disk drives were recently reported in a computer magazine. Three variables were recorded for each hard disk drive:
$y =$ Retail PRICE (measured in dollars)
$x _ { 1 } =$ Microprocessor SPEED (measured in megahertz)
(Values in sample range from 10 to 40 )
$x _ { 2 } =$ CHIP size (measured in computer processing units)
(Values in sample range from 286 to 486 )

a first-order regression model was fit to the data. Part of the printout follows:

$\begin{array}{rrrrrrrrr}
\hline &&&\text { Dep Var}&\text { Predict}&\text { Std Err }&\text {Lower 95\% }&\text {Upper 95\% }\
\text { OBS}& \text {SPEED}&\text {  CHIP } & \text { PRICE } & \text { Value } & \text { Predict } & \text { Predict } & \text { Predict } & \text { Residual } \
1 & 33 & 286 & 5099.0 & 4464.9 & 260.768 & 3942.7 & 4987.1 & 634.1 \
\hline
\end{array}$

Interpret the interval given in the printout.

A) We are $95 \%$ confident that the price of a single hard drive falls between $\$ 3,943$ and $\$ 4,987$.
B) We are $95 \%$ confident that the average price of all hard drives falls between $\$ 3,943$ and $\$ 4,987$.
C) We are $95 \%$ confident that the price of a single hard drive with 33 megahertz speed and 386 CPU falls between $\$ 3,943$ and $\$ 4,987$.
D) We are $95 \%$ confident that the average price of all hard drives with 33 megahertz speed and 386 CPU falls between $\$ 3,943$ and $\$ 4,987$.

Accepted Answer

The interval provided is a prediction interval for a single observation, indicating the range within which we expect the price of a single hard drive with the specified characteristics to fall.

Question 8

One of three surfaces is produced by a complete second-order model with two quantitative independent variables: a paraboloid that opens upward, a paraboloid that opens downward, or a saddle-shaped surface.

Accepted Answer

A complete second-order model with two quantitative independent variables can produce a paraboloid that opens upward, a paraboloid that opens downward, or a saddle-shaped surface.

Question 9

As part of a study at a large university, data were collected on $n = 224$ freshmen computer science (CS) majors in a particular year. The researchers were interested in modeling $y$, a student's grade point average (GPA) after three semesters, as a function of the following independent variables (recorded at the time the students enrolled in the university):
$x _ { 1 } =$ average high school grade in mathematics (HSM)
$x _ { 2 } =$ average high school grade in science (HSS)
$x _ { 3 } =$ average high school grade in English (HSE)
$x _ { 4 } =$ SAT mathematics score (SATM)
$x _ { 5 } =$ SAT verbal score (SATV)

A first-order model was fit to data with the following results:

$\begin{array}{lrrrrr}
\hline \text { SOURCE } & \text { DF } & \text { SS } & \text { MS } & \text { F VALUE } & \text { PROB > F } \
\text { MODEL } & 5 & 28.64 & 5.73 & 11.69 & .0001 \
\text { ERROR } & 218 & 106.82 & 0.49 & & \
\text { TOTAL } & 223 & 135.46 & & &
\end{array}$

$\begin{array}{lccc}
\text { ROOT MSE } & 0.700 & \text { R-SQUARE } & 0.211 \
\text { DEP MEAN } & 4.635 & \text { ADJR-SQ } & 0.193
\end{array}$
 $\begin{array}{l}
\begin{array} { l r r r r } 
& \text {PARAMETER }& \text {STANDARD}& \text {T FOR 0:}\
 \text {VARIABLES}&\text { ESTIMATE } & \text { ERROR } & \text { PARAMETER } = 0 & \text { PROB } > | T |  \ 
\
\text { INTERCEPT } & 2.327 & 0.039 & 5.817 & 0.0001 \
\text { X1 (HSM) } & 0.146 & 0.037 & 3.718 & 0.0003 \
\text { X2 (HSS) } & 0.036 & 0.038 & 0.950 & 0.3432 \
\text { X3 (HSE) } & 0.055 & 0.040 & 1.397 & 0.1637 \
\text { X4 (SATM) } & 0.00094 & 0.00068 & 1.376 & 0.1702 \
\text { X5 (SATV) } & - 0.00041 & 0.00059 & - 0.689 & 0.4915 \
\hline
\end{array}
\end{array}$

Interpret the value under the column heading $P R O B > F$.

A) Accept $H _ { 0 }$ (at $\alpha = .01$ ); at least one of the $\beta$-coefficients in the first-order model is equal to 0 .
B) Over $99 \%$ of the variation in GPAs can be explained by the model.
C) There is insufficient evidence (at $\alpha = .01$ ) to conclude that the first-order model is statistically useful for predicting GPA.
D) There is sufficient evidence (at $\alpha = .01$ ) to conclude that the first-order model is statistically useful for predicting GPA.

Accepted Answer

The $P$-value for the model is 0.0001, which is less than $\alpha = 0.01$, indicating that the model is statistically significant and useful for predicting GPA.

Question 10

A certain type of rare gem serves as a status symbol for many of its owners. In theory, for low prices, the demand decreases as the price of the gem increases. However, experts hypothesize that when the gem is valued at very high prices, the demand increases with price due to the status the owners believe they gain by obtaining the gem. Thus, the model proposed to best explain the demand for the gem by its price is the quadratic model $$E ( y ) = \beta _ { 0 } + \beta _ { 1 } x + \beta _ { 2 } x ^ { 2 }$$ where $y =$ Demand (in thousands) and $x =$ Retail price per carat (dollars). This model was fit to data collected for a sample of 12 rare gems. If the experts are correct in their assumptions about the relationship between price and demand, which of the following should be true? A) $\beta 1 < 0$ B) $\beta _ { 2 } < 0$ C) $\beta _ { 2 } > 0$ D) $\beta 1 > 0$

Accepted Answer

When the price is very high, the demand increases with price. This means that the quadratic term $\beta _ { 2 } $ must be positive.

Question 11

In the quadratic model $E ( y ) = \beta _ { 0 } + \beta _ { 1 } x + \beta _ { 2 } x ^ { 2 } , \text { a negative value of } \beta _ { 1 }$ indicates downward
concavity.

Accepted Answer

A negative value of $\beta_1$ affects the slope of the linear term, not the concavity. Concavity is determined by the sign of $\beta_2$.

Quiz 12: Multiple Regression and Model Building

A regression residual is the difference between an observed y value and its corresponding predicted value.

The stepwise regression model should not be used as the final model for predicting y.

The independent variables x1 and x2 interact when the effect on E(y) of a change in x1 depends on x2.

The rejection of the null hypothesis in a global F-test means that the model is the best model for providing reliable estimates and predictions.

One of three surfaces is produced by a complete second-order model with two quantitative independent variables: a paraboloid that opens upward, a paraboloid that opens downward, or a saddle-shaped surface.

In the quadratic model E(y)=β0+β1x+β2x2, a negative value of β1E ( y ) = \beta _ { 0 } + \beta _ { 1 } x + \beta _ { 2 } x ^ { 2 } , \text { a negative value of } \beta _ { 1 }E(y)=β0​+β1​x+β2​x2, a negative value of β1​ indicates downward concavity.

The independent variables x₁ and x₂ interact when the effect on E(y) of a change in x₁ depends on x₂.

In the quadratic model $E ( y ) = \beta _ { 0 } + \beta _ { 1 } x + \beta _ { 2 } x ^ { 2 } , \text { a negative value of } \beta _ { 1 }$ indicates downward concavity.