Deck 16: B: Simple Linear Regression and Correlation

Question 1

The variance of the error variable $\sigma _ { z } ^ { 2 }$ is required to be constant.When this requirement is violated,the condition is called heteroscedasticity.

Answer 1

True

Question 2

Error terms that are correlated over time are said to be ____________________.

Answer 2

autocorrelated
serially correlated

Question 3

If the variance of the errors is constant for each predicted y value,the condition is called ____________________.

Answer 3

homoscedasticity

Question 4

When the variance $\sigma _ { z } ^ { 2 }$ of the error variable $\varepsilon$ is a constant no matter what the value of x is,this condition is called:

A) heterocausality.
B) homoscedasticity.
C) homocausality.
D) heteroscedasticity.

Question 5

Error terms that are autocorrelated ____________________ (are/are not)independent.

Question 6

We check for normality by drawing a(n)____________________ of the residuals.

Question 7

The plot of residuals vs.predicted values should show no patterns if the conditions of a regression analysis are met.

Question 8

One method of diagnosing heteroscedasticity is to plot the residuals against the predicted values of y,then look for a change in the spread of the plotted values.

Question 9

We standardize residuals by subtracting their mean and dividing by their variance.

Question 10

If you take the residuals,subtract their mean and divide by their standard deviation,the result is called the ____________________ residuals.

Question 11

We check for normality by drawing a pie chart of the residuals.

Question 12

The least squares method requires that the variance $\sigma _ { z } ^ { 2 }$ of the error variable $\varepsilon$ is a constant no matter what the value of x is.When this requirement is violated,the condition is called:

A) heteroscedasticity.
B) homoscedasticity.
C) influential observation.
D) non-independence of $\varepsilon$ .

Question 13

If the plot of the residuals vs.the predicted values resembles a straight line with non-zero slope,then the regression line fits well.

Question 14

An outlier is an observation that is unusually small or unusually large.

Question 15

When the error variable does not have constant variance,this condition is called ____________________.

Question 16

Data that exhibit an autocorrelation effect violate the regression assumption of independence.

Question 17

If the plot of the residuals is fan shaped,which assumption of regression analysis (if any)is violated?

A) No assumptions are violated.
B) Independence of errors
C) Homoscedasticity
D) Normality

Question 18

The standardized residual is defined as:

A) residual multiplied by the square root of the standard error of estimate.
B) residual multiplied by the standard error of estimate.
C) residual divided by the square of the standard error of estimate.
D) residual divided by the standard error of estimate.

Question 19

The spread in the residuals should increase as the predicted value of y increases.

Question 20

The variance of the error variable $\sigma _ { z } ^ { 2 }$ is required to be constant.When this requirement is satisfied,the condition is called homoscedasticity.

Question 21

Game Winnings & Education
An ardent fan of television game shows has observed that,in general,the more educated the contestant,the less money he or she wins.To test her belief she gathers data about the last eight winners of her favorite game show.She records their winnings in dollars and the number of years of education.The results are as follows.
$$\begin{array} { | c | c | c | } \hline \text { Contestant } & \text { Years of Education } & \text { Winnings } \\\hline 1 & 11 & 750 \\2 & 15 & 400 \\3 & 12 & 600 \\4 & 16 & 350 \\5 & 11 & 800 \\0 & 16 & 300 \\7 & 13 & 650 \\8 & 14 & 400 \\\hline\end{array}$$

-{Game Winnings & Education Narrative} Use the predicted and actual values of y to calculate the residuals.

Question 22

Marc Anthony Concert
At a recent Marc Anthony concert,a survey was conducted that asked a random sample of 20 people their age and how many concerts they have attended since the first of the year.The following data were collected:
An Excel output follows:
{Marc Anthony Concert Narrative} Plot the residuals against the predicted values .

Question 23

Marc Anthony Concert
At a recent Marc Anthony concert,a survey was conducted that asked a random sample of 20 people their age and how many concerts they have attended since the first of the year.The following data were collected:
An Excel output follows:
{Marc Anthony Concert Narrative} Does it appear that the errors are normally distributed? Explain.

Question 24

Game Winnings & Education
An ardent fan of television game shows has observed that,in general,the more educated the contestant,the less money he or she wins.To test her belief she gathers data about the last eight winners of her favorite game show.She records their winnings in dollars and the number of years of education.The results are as follows.

{Game Winnings & Education Narrative} Use the regression equation to determine the predicted values of y.

Question 25

Marc Anthony Concert
At a recent Marc Anthony concert,a survey was conducted that asked a random sample of 20 people their age and how many concerts they have attended since the first of the year.The following data were collected:
$$\begin{array}{l}\begin{array} { | l | c c c c c c c c c c | } \hline \text { Age } & 62 & 57 & 40 & 49 & 67 & 54 & 43 & 65 & 54 & 41 \\\hline \text { Number af Concerts } & 6 & 5 & 4 & 3 & 5 & 5 & 2 & 6 & 3 & 1 \\\hline\end{array}\\\\\begin{array} { | l | c c c c c c c c c c | } \hline \text { Age } & 44 & 48 & 55 & 60 & 59 & 63 & 69 & 40 & 38 & 52 \\\hline \text { Number of Concerts } & 3 & 2 & 4 & 5 & 4 & 5 & 4 & 2 & 1 & 3 \\\hline\end{array}\end{array}$$ An Excel output follows:

-{Marc Anthony Concert Narrative} Use the residuals to compute the standardized residuals.

Question 26

Game Winnings & Education
An ardent fan of television game shows has observed that,in general,the more educated the contestant,the less money he or she wins.To test her belief she gathers data about the last eight winners of her favorite game show.She records their winnings in dollars and the number of years of education.The results are as follows.

{Game Winnings & Education Narrative} Identify possible outliers.

Question 27

Game Winnings & Education
An ardent fan of television game shows has observed that,in general,the more educated the contestant,the less money he or she wins.To test her belief she gathers data about the last eight winners of her favorite game show.She records their winnings in dollars and the number of years of education.The results are as follows.
$$\begin{array} { | c | c | c | } \hline \text { Contestant } & \text { Years of Education } & \text { Winnings } \\\hline 1 & 11 & 750 \\2 & 15 & 400 \\3 & 12 & 600 \\4 & 16 & 350 \\5 & 11 & 800 \\0 & 16 & 300 \\7 & 13 & 650 \\8 & 14 & 400 \\\hline\end{array}$$

-{Game Winnings & Education Narrative} Compute the standardized residuals.

Question 28

Marc Anthony Concert
At a recent Marc Anthony concert,a survey was conducted that asked a random sample of 20 people their age and how many concerts they have attended since the first of the year.The following data were collected:
An Excel output follows:
{Marc Anthony Concert Narrative} Does it appear that heteroscedasticity is a problem? Explain.

Question 29

Game Winnings & Education
An ardent fan of television game shows has observed that,in general,the more educated the contestant,the less money he or she wins.To test her belief she gathers data about the last eight winners of her favorite game show.She records their winnings in dollars and the number of years of education.The results are as follows.

{Game Winnings & Education Narrative} Plot the residuals against the predicted values .What does the graph tell you?

Question 30

If a single point has a large impact on the equation of the regression line,it is called a(n)____________________ point.

Question 31

Marc Anthony Concert
At a recent Marc Anthony concert,a survey was conducted that asked a random sample of 20 people their age and how many concerts they have attended since the first of the year.The following data were collected:
An Excel output follows:
{Marc Anthony Concert Narrative} Identify possible outliers.

Question 32

Comedy Shows Revenues
A financier whose specialty is investing in comedy shows has observed that,in general,shows with "big-name" stars seem to generate more revenue than those shows whose stars are less well known.To examine his belief he records the gross revenue and the payment (in $ millions)given to the two highest-paid comedians in the show for ten recently staged shows.

{Comedy Shows Revenues Narrative} Use the regression equation to determine the predicted values of y.

Question 33

We can often detect autocorrelation by graphing the residuals against ____________________.

Question 34

Marc Anthony Concert
At a recent Marc Anthony concert,a survey was conducted that asked a random sample of 20 people their age and how many concerts they have attended since the first of the year.The following data were collected:
An Excel output follows:
{Marc Anthony Concert Narrative} Use the regression equation to determine the predicted values of y.

Question 35

Marc Anthony Concert
At a recent Marc Anthony concert,a survey was conducted that asked a random sample of 20 people their age and how many concerts they have attended since the first of the year.The following data were collected:
An Excel output follows:
{Marc Anthony Concert Narrative} Draw a histogram of the residuals.

Question 36

Marc Anthony Concert
At a recent Marc Anthony concert,a survey was conducted that asked a random sample of 20 people their age and how many concerts they have attended since the first of the year.The following data were collected:
An Excel output follows:
{Marc Anthony Concert Narrative} Use the predicted values and the actual values of y to calculate the residuals.

Question 37

A(n)____________________ is an observation that is unusually small or large.

Question 38

Comedy Shows Revenues
A financier whose specialty is investing in comedy shows has observed that,in general,shows with "big-name" stars seem to generate more revenue than those shows whose stars are less well known.To examine his belief he records the gross revenue and the payment (in $ millions)given to the two highest-paid comedians in the show for ten recently staged shows.
$$\begin{array} { | c | c | c | } \hline \text { Show } & \begin{array} { c } \text { Cost of Twa Highest } \\\text { Paid Comedian (\$mil) }\end{array} & \begin{array} { c } \text { Gross Revenue } \\( \mathbf { \$ m i l } )\end{array} \\\hline 1 & 5.3 & 48 \\2 & 7.2 & 65 \\3 & 1.3 & 18 \\4 & 1.8 & 20 \\5 & 3.5 & 31 \\6 & 2.6 & 26 \\7 & 8.0 & 73 \\8 & 2.4 & 23 \\9 & 4.5 & 39 \\10 & 6.7 & 58 \\\hline\end{array}$$

-{Comedy Shows Revenues Narrative} Use the predicted and actual values of y to calculate the residuals.

Question 39

Comedy Shows Revenues
A financier whose specialty is investing in comedy shows has observed that,in general,shows with "big-name" stars seem to generate more revenue than those shows whose stars are less well known.To examine his belief he records the gross revenue and the payment (in $ millions)given to the two highest-paid comedians in the show for ten recently staged shows.

{Comedy Shows Revenues Narrative} Plot the residuals against the predicted values of y.What does the graph tell you?

Question 40

Oil Quality and Price
Quality of oil is measured in API gravity degrees--the higher the degrees API,the higher the quality.The table shown below is produced by an expert in the field who believes that there is a relationship between quality and price per barrel.
A partial Minitab output follows:

{Oil Quality and Price Narrative} Use the regression equation to determine the predicted values of y.

Question 41

Oil Quality and Price
Quality of oil is measured in API gravity degrees--the higher the degrees API,the higher the quality.The table shown below is produced by an expert in the field who believes that there is a relationship between quality and price per barrel.
$$\begin{array} { | c | c | } \hline \text { Oi degrees API } & \text { Price per barrel (in \$) } \\\hline 27.0 & 12.02 \\28.5 & 12.04 \\30.8 & 12.32 \\31.3 & 12.27 \\31.9 & 12.49 \\34.5 & 12.70 \\34.0 & 12.80 \\34.7 & 13.00 \\37.0 & 13.00 \\41.0 & 13.17 \\41.0 & 13.19 \\38.8 & 13.22 \\39.3 & 13.27 \\\hline\end{array}$$ A partial Minitab output follows:
$$\begin{array}{l}\text { Dascriptive atafistics }\\\begin{array} { l l l l l } \text { Variable } & \mathrm { N } & \text { Mean } & \text { StDev } & \text { SE Mean } \\\text { Degrees } & 13 & 34.60 & 4.613 & 1.280 \\\text { Price } & 13 & 1270 & 0.757 & 0.127\end{array}\end{array}$$ $$\begin{array}{l}\begin{array} { l l } \text {Covariances }\\&\text { Degeres } & \text { Price } \\\text { Degeres }&21.281667 & \\\text { Price } &2.026750 & 0.208933\end{array}\end{array}$$ $$\begin{array}{l}\begin{array} { l l l r } \text { Rederatian Antalyis } & & \\\text { predictor } & \text { Caef } & \text { StDev } & \text { T }&\text { P} \\\text { Cunstant } & 9.4349 & 0.2867 & 32.91&0.000 \\\text { Degrees } & 0.095235 &0 .008220 & 11.59&0.000\end{array}\\\\S =0 .1314 \quad R - S q = 92.46 \% \quad R - S q ( a dj ) = 91.7 \%\end{array}$$ $$\begin{array}{l}\text { Analysis of Variance }\\\begin{array} { l l r r r } \text { Surce } & \text { DF } & \text { SS } & \text { MS } & \text { F } &\text {P}\\\text { Regeression } & 1 & 2.3162 & 2.3162 & 134.24 &0.000\\\text { Residual Error } & 11 & 0.1898 & 0.0173 & \\\text { Total } & 12 & 2.5060 & &\end{array}\end{array}$$

-{Oil Quality and Price Narrative} Use the predicted values and the actual values of y to calculate the residuals.

Question 42

Oil Quality and Price
Quality of oil is measured in API gravity degrees--the higher the degrees API,the higher the quality.The table shown below is produced by an expert in the field who believes that there is a relationship between quality and price per barrel.
A partial Minitab output follows:

{Oil Quality and Price Narrative} Plot the residuals against the predicted values .

Question 43

{Oil Quality and Price Narrative} Identify possible outliers.

Question 44

Oil Quality and Price
Quality of oil is measured in API gravity degrees--the higher the degrees API,the higher the quality.The table shown below is produced by an expert in the field who believes that there is a relationship between quality and price per barrel.
A partial Minitab output follows:

{Oil Quality and Price Narrative} Does it appear that heteroscedasticity is a problem? Explain.

Question 45

Oil Quality and Price
Quality of oil is measured in API gravity degrees--the higher the degrees API,the higher the quality.The table shown below is produced by an expert in the field who believes that there is a relationship between quality and price per barrel.
$$\begin{array} { | c | c | } \hline \text { Oi degrees API } & \text { Price per barrel (in \$) } \\\hline 27.0 & 12.02 \\28.5 & 12.04 \\30.8 & 12.32 \\31.3 & 12.27 \\31.9 & 12.49 \\34.5 & 12.70 \\34.0 & 12.80 \\34.7 & 13.00 \\37.0 & 13.00 \\41.0 & 13.17 \\41.0 & 13.19 \\38.8 & 13.22 \\39.3 & 13.27 \\\hline\end{array}$$ A partial Minitab output follows:
$$\begin{array}{l}\text { Dascriptive atafistics }\\\begin{array} { l l l l l } \text { Variable } & \mathrm { N } & \text { Mean } & \text { StDev } & \text { SE Mean } \\\text { Degrees } & 13 & 34.60 & 4.613 & 1.280 \\\text { Price } & 13 & 1270 & 0.757 & 0.127\end{array}\end{array}$$ $$\begin{array}{l}\begin{array} { l l } \text {Covariances }\\&\text { Degeres } & \text { Price } \\\text { Degeres }&21.281667 & \\\text { Price } &2.026750 & 0.208933\end{array}\end{array}$$ $$\begin{array}{l}\begin{array} { l l l r } \text { Rederatian Antalyis } & & \\\text { predictor } & \text { Caef } & \text { StDev } & \text { T }&\text { P} \\\text { Cunstant } & 9.4349 & 0.2867 & 32.91&0.000 \\\text { Degrees } & 0.095235 &0 .008220 & 11.59&0.000\end{array}\\\\S =0 .1314 \quad R - S q = 92.46 \% \quad R - S q ( a dj ) = 91.7 \%\end{array}$$ $$\begin{array}{l}\text { Analysis of Variance }\\\begin{array} { l l r r r } \text { Surce } & \text { DF } & \text { SS } & \text { MS } & \text { F } &\text {P}\\\text { Regeression } & 1 & 2.3162 & 2.3162 & 134.24 &0.000\\\text { Residual Error } & 11 & 0.1898 & 0.0173 & \\\text { Total } & 12 & 2.5060 & &\end{array}\end{array}$$

-{Oil Quality and Price Narrative} Use the residuals to compute the standardized residuals.

Question 46

Oil Quality and Price
Quality of oil is measured in API gravity degrees--the higher the degrees API,the higher the quality.The table shown below is produced by an expert in the field who believes that there is a relationship between quality and price per barrel.
A partial Minitab output follows:

{Oil Quality and Price Narrative} Does it appear that the errors are normally distributed? Explain.

Question 47

Oil Quality and Price
Quality of oil is measured in API gravity degrees--the higher the degrees API,the higher the quality.The table shown below is produced by an expert in the field who believes that there is a relationship between quality and price per barrel.
A partial Minitab output follows:

{Oil Quality and Price Narrative} Draw a histogram of the residuals.