Question 1

Mistakes. Describe the mistake made in the following analyses:
a. Ten teachers compute their average test scores for all their students. Then the
superintendent collects their data and finds the school average. He repeats this process for
eight different schools and finds a positive correlation between the age of the school
average age of the teachers at a school and their average score.
b. The mayor of a city is concerned that the population of the city is growing faster than
revenue. He calculates that over the last 5 years, the year and the size of the city have a
R-sq of 95.7%. With such a high value, the mayor confidently predicts the population for
the next three years of fiscal planning.

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The answer of Mistakes. Describe the mistake made in the...

Question 2

Engineers want to know what factors are associated with gas mileage. The regression
below predicts the average miles per gallon (MPG) for 82 cars using their engine
horsepower (HP) and weight (WT, in 100's of pounds). Dependent variable is $\mathrm { MPG }$
R squared $= 82.4 \% \quad$ R squared (adjusted) $= 81.9 \%$
$\mathrm { s } = 4.255$ with $82 - 3 = 79$ degrees of freedom
\begin{tabular} { l c l c c } 
\multicolumn{1}{c} { Source } & Sum of Squares & df & Mean Square & Fratio \
Regression & $6676.84$ & 2 & $3338.42$ & 184 \
Residual & $1430.62$ & 79 & $18.1091$ &
\end{tabular}
$\begin{array} { l l l l r } \text { Variable } & \text { Coefficient } & \text { SE(Coeff) } & \text { t-ratio } & \text { p-value } \ \text { Constant } & 66.855 & 2.079 & 32.2 & < 0.0001 \ \text { HP } & - 0.0209708 & 0.015 & - 1.4 & 0.1661 \ \text { WT } & - 0.990369 & 0.1047 & - 9.45 & < 0.0001 \end{array}$ 
a. Write down the regression equation.
b. Write down the hypotheses for the test of the coefficient of horsepower. Conduct the test
and explain your conclusion in the context of this problem.
c. Write down the hypotheses for the test of the coefficient of weight. Conduct the test and
explain your conclusion in the context of this problem.
d. Explain the meaning of the coefficient of weight in the context of this problem.
e. Explain the meaning of the intercept of this regression in the context of this problem.
f. Compute the predicted gas mileage of a 3500 pound car with a 150 horsepower engine.
The plots are MPG vs. HP, MPG vs. WT, residuals vs. predicted values, and a normal
probability plot of residuals.  
g. Check the conditions of this regression and comment on whether they are satisfied.

Accepted Answer

The answer of Engineers want to know what factors are...

Question 3

Here is the regression with both Latitude and Longitude as predictors: Dependent variable is: JanTemp
$R$ squared $= 74.1 \% \quad R$ squared (adjusted) $= 73.1 \%$
$\mathrm { s } = 7.000$ with $55 - 3 = 52$ degrees of freedom
$\begin{array} { l l l l l } \text { Source } & \text { Sum of Squares } & \text { df } & \text { Mean Square } & \text { F-ratio } \ \text { Regression } & 7277.18 & 2 & 3638.59 & 74.2 \ \text { Residual } & 2548.35 & 52 & 49.0067 & \ & & & & \ \text { Variable } & \text { Coefficient } & \text { SE(Coeff) } & \text { t-ratio } & \text { P-value } \ \text { Intercept } & 98.5620 & 8.473 & 11.6 & \leq 0.0001 \ \text { Lat } & - 2.16286 & 0.1776 & - 12.2 & \leq 0.0001 \ \text { Long } & 0.134471 & 0.0641 & 2.10 & 0.0407 \end{array}$ 
The coefficient of Long in this regression differs from the coefficient of Long in the simple
regression of JanTemp on Long. What is the meaning of the coefficient of Long in this
regression? Are you confident ( $$( \text { at } \alpha = 0.05 )$$ 
that the coefficient is not zero? Why or why
not?

Accepted Answer

The answer of Here is the regression with both Latitude...

Question 4

Here is a partial regression plot for the coefficient of Long in the regression with a least
squares regression line added to the display:  
What is the slope of the line on this display? Does the display suggest that this slope
adequately summarizes the effect of longitude in the regression? Why/Why not?

Accepted Answer

The answer of Here is a partial regression plot for...

Question 5

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Of the 23 first year male students at State U. admitted from Jim Thorpe High School, 8 were offered baseball scholarships and
7 were offered football scholarships. The University admissions committee looked at the students' composite ACT scores
(shown in the tabl, wondering if the University was lowering their standards for athletes. Assuming that this group of
students is representative of all admitted students, what do you think? 
-Test an appropriate hypothesis and state your conclusion.

Accepted Answer

The answer of SHORT ANSWER. Write the word or phrase...

Question 6

For a class project, students tested four different brands of laundry detergent (1, 2, 3, 4) in
three different water temperatures (hot, warm, cold) to see whether their were any
differences in how well the detergents could clean clothes. The students took 36 identical
pieces of cloth and made them dirty by staining them with coffee, dirt, and grass. The 36
pieces were randomly assigned to the 12 combinations of detergent and temperature so
that each combination had 3 replicates. After washing, the students rated how clean the
clothes were from 0 (no change) to 20 (completely spotless). The two factor ANOVA table
is shown below along with an interaction plot and residual plots. $$\begin{array} { | l r r r r r | } 
\hline \text { Source } & \text { df } & \begin{array} { c } 
\text { Sums of } \
\text { Squares }
\end{array} & \begin{array} { l } 
\text { Mean } \
\text { Square }
\end{array} & \text { F-ratio } & \text { P-value } \
\hline \text { Detergent } & 3 & 38.972 & 12.9907 & 3.966 & 0.0171 \
\text { Temp } & 2 & 181.056 & 90.5278 & 27.634 & < 0.0001 \
\text { Error } & 30 & 98.278 & 3.2759 & & \
\text { Total } & 35 & 318.306 & & & \
\hline
\end{array}$$ 
  
a. Write the hypotheses tested by the Detergent F-ratio. Test the hypotheses and explain
your conclusion in the context of the problem.
b. Write the hypotheses tested by the Temp F-ratio. Test the hypotheses and explain your
conclusion in the context of the problem.
c. Check the conditions required for the ANOVA analysis.

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The answer of For a class project, students tested four...

Question 7

It is possible that the distance that a city is from the ocean could affect its average January
low temperature. Coast gives an approximate distance of each city from the East Coast or
West Coast (whichever is nearer). Including it in the regression yields the following
regression table: Dependent variable is:JanTemp
R squared $= 87.6 \% \quad$ R squared (adjusted) $= 86.9 \%$ $\mathrm { s } = 4.878$ with $55 - 4 = 51$ degrees of freedom
$\begin{array} { l c c c c } \text { Source } & \text { Sum of Squares } & \text { df } & \text { Mean Square } & \text { F-ratio } \ \text { Regression } 8611.86 & 3 & 2870.62 & 121 \ \text { Residual } & 1213.67 & 51 & 23.7974 & \end{array}\$
$\begin{array} { l c l l r } \text { Variable } & \text { Coefficient } & \text { SE(Coeff) } & \text { t-ratio } & \text { P-value } \ \text { Intercept } & 111.878 & 6.167 & 18.1 & \leq 0.0001 \ \text { Lat } & - 2.47722 & 0.1307 & - 19.0 & \leq 0.0001 \ \text { Long } & 0.221997 & 0.0462 & 4.81 & \leq 0.0001 \ \text { Coast } & - 0.674929 & 0.0901 & - 7.49 & \leq 0.0001 \end{array}$ 
And here is a scatterplot of the residuals: 
$$\text { Write a report on this regression. Interpret the coefficients and } R ^ { 2 } \text {. Are the conditions met? }$$

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The answer of It is possible that the distance that...

Question 8

What is the predicted number of skiers for a Saturday with a temperature of 40° F. and a
snow cover of 25 inches?

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The answer of What is the predicted number of skiers...

Question 9

Shrimp From 1982 to 1990, there was a decrease in the number of white shrimp harvested
from the Galveston Bay. Here is the regression analysis and a residual plot. The year has
been shortened to two digits (82, 83…) and the dependent variable is the number of shrimp
collected per hour. Dependent Variable: Shrimp/hour
$$\begin{array} { l } 
\text { R-sq } = 0.9496342 \
\mathrm {~s} : 6.0232354
\end{array}$$
$\begin{array} { l r r } \text { Parameter } & \text { Estimate } & \text { Std. Err. } \ \text { constant } & 816.71111 & 66.903419 \ \text { year } & - 8.9333333 & 0.77759635 \end{array}$

a. Write the regression equation and define your variables.
b. Find the correlation coefficient and interpret it in context.
c. Interpret the value of the slope in context.
d. In 1991, the shrimp production rebounded (in part due to the effects of El Nino) to 81
shrimp/hour. Find the value of this residual.
e. The prediction for 1991 was very inaccurate. What name do statisticians give to this kind
of prediction error?

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The answer of Shrimp From 1982 to 1990, there was...

Question 10

If the diameter is increased from 4 inches to 14 inches, the predicted number of cheerios will
Increase by about…
A) 72
B) 10
C) 21
D) none of these
E) 51

Accepted Answer

The number of Cheerios is proportional to the square of the diameter. So, if the diameter is increased by a factor of 14/4 = 3.5, the number of Cheerios will increase by a factor of 3.5^2 = 12.25. This is an increase of about 51 Cheerios.

Question 11

Here is the corresponding regression table: Dependent variable is: JanTemp
R squared $= 71.9 \% \quad$ R squared (adjusted) $= 71.3 \%$
$\mathrm { s } = 7.222$ with $55 - 2 = 53$ degrees of freedom
$\begin{array} { l l l l l } \text { Source } & \text { Sum of Squares } & \text { df } & \text { Mean Square } & \text { F-ratio } \ \text { Regression } & 7061.32 & 1 & 7061.32 & 135 \ \text { Residual } & 2764.21 & 53 & 52.1549 & \ & & & & \ \text { Variable } & \text { Coefficient } & \text { SE(Coeff) } & \text { t-ratio } & \text { P-value } \ \text { Intercept } & 108.805 & 7.146 & 15.2 & \leq 0.0001 \ \text { Lat } & - 2.11114 & 0.1814 & - 11.6 & \leq 0.0001 \end{array}$ 
Write a brief report based on this regression. Explain in words and numbers what this
equation says about the relationship between average January low temperature and
latitude. Discuss the R2 value and t-ratios.

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The answer of Here is the corresponding regression table: Dependent...

Question 12

Are the two sports teams mean ACT scores different?

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The answer of Are the two sports teams mean ACT...

Question 13

Check the conditions for the regression and comment on whether or not they are satisfied.

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The answer of Check the conditions for the regression and...

Question 14

Which of the explanatory variables appear to be associated with the number of skiers, and
which do not? Explain how you reached your conclusion.

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The answer of Which of the explanatory variables appear to...

Question 15

A student wants to build a paper airplane that gets maximum flight distance. She tries
three ways of bending the wing (down, flat, and up) and two levels of nose weight (no and yes
- a paper clip). She randomizes the 12 runs (each condition replicated twice). The analysis
of variance for the 12 runs is shown in the table below along with an interaction plot and
tables of the mean distance for the different wing bends and weights. $$\begin{array}{l}
\begin{array} { | l r r r r r | } 
\hline \text { Source } & \text { df } & \begin{array} { l } 
\text { Sums of } \
\text { Squares }
\end{array} & \begin{array} { l } 
\text { Mean } \
\text { Square }
\end{array} & \text { F-ratio } & \text { P-value } \
\hline \text { Wing Bend } & 2 & 13565.2 & 6782.58 & 152.7 & < 0.0001 \
\text { Weight } & 1 & 6768.75 & 6768.75 & 152.39 & < 0.0001 \
\text { Interaction } & 2 & 186.5 & 93.25 & 2.0994 & 0.2036 \
\text { Error } & 6 & 266.5 & 44.4167 & & \
\text { Total } & 11 & 20786.9 & & & \
\hline
\end{array}\\
\begin{array} { | c c | } 
\hline \text { Wing Bend } & \begin{array} { c } 
\text { Expected } \
\text { Mean }
\end{array} \
\hline \text { Down } & 145.0 \
\text { Flat } & 182.5 \
\text { Up } & 227.2 \
\hline
\end{array}\\
\begin{array} { | c c | } 
\hline \text { Weight } & \begin{array} { c } 
\text { Expected } \
\text { Mean }
\end{array} \
\hline \text { No } & 161.2 \
\text { Yes } & 208.7 \
\hline
\end{array}
\end{array}$$

a. Does an additive model seem adequate? Explain.
b. Write a report on this analysis of the data. Include any recommendations you would
give the student on designing the plane.

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The answer of A student wants to build a paper...

Question 16

straight so it is reasonable to assume that the regression errors follow the Normal
model.
-If you think that the temperature might affect attendance differently on weekends than on
weekdays, how would you change the regression to test this?

Accepted Answer

The answer of straight so it is reasonable to assume...

Question 17

Here are data about the average January low temperature in cities in the United States, and factors that might allow us to
predict temperature. The data, available for 55 cities, include:
JANTEMP Mean January low temperature (degrees F.)
LAT Latitude (degrees of latitud
LONG Longitude (degrees of longitud
ALTITUDE Altitude at the airport (where the temperatures have been recorded, in feet above sea level)
COAST Approximate distance from the nearest seacoast (east or west, in degrees longitud
We will attempt to make a regression model to help account for mean January temperature and to understand the effects of the
various predictors.
At each step of the analysis you may assume that things learned earlier in the process are known.
Units Note: The "degrees" of temperature, given here on the Fahrenheit scale, have only coincidental language relationship to the
"degrees" of longitude and latitude. The geographic "degrees" are based on modeling the Earth as a sphere and dividing it up into 360
degrees for a full circle. Thus 180 degrees of longitude is halfway around the world from Greenwich, England (0°) and Latitude
increases from 0 degrees at the Equator to 90 degrees of (North) latitude at the North Pole. 
-First, we consider the relationship between temperature and latitude. This seems to be the
obvious first choice; everybody knows that northern (high latitude) cities tend to be colder
in January than southern (lower latitude) cities. Here is the scatterplot:  
Describe what you see in this scatterplot in a sentence or two. Which of the regression
assumptions for the regression of Jantemp on Lat can you check with this plot? State them
and indicate whether you think they seem to be satisfied.

Accepted Answer

The answer of Here are data about the average January...

Question 18

Students A growing school district tracks the student population growth over the years
from 2008 to 2013. Here are the regression results and a residual plot. students $= 119.53 + 172.03$ year
Sample size: 6
$\mathrm { R } - \mathrm { sq } = 0.987$

a. Explain why despite a high $\mathrm { R } - \mathrm { sq }$, this regression is not a successful model.
To linearize the data, the $\log$ (base 10 ) was taken of the student population. Here are the results.
Dependent Variable: $\log$ (students)
Sample size: 6
$$\mathrm { R } - \mathrm { sq } = 0.994$$
$\begin{array} { l r r } \text { Parameter } & \text { Estimate } & \text { Std. Err. } \ \text { constant } & 2.871 & 0.0162 \ \text { year } & 0.0389 & 0.00152 \end{array}$ 
  
b. Describe the success of the linearization.
c. Interpret R-sq in the context of this problem.
d. Predict the student population in 2014.

Accepted Answer

The answer of Students A growing school district tracks the...

Question 19

Compute a 95% confidence interval for the slope of the variable Weekend, and explain the
meaning of the interval in the context of the problem.

Accepted Answer

The answer of Compute a 95% confidence interval for the...

Question 20

Now, consider longitude. Should the longitude of a city have an influence on average
January low temperature? Here is the regression: Dependent variable is: JanTemp
$\mathrm { R }$ squared $= 0.1 \% \quad \mathrm { R }$ squared (adjusted) $= - 1.8 \%$
$s = 13.61$ with $55 - 2 = 53$ degrees of freedom
$\begin{array} { l l l l l } \text { Source } & \text { Sum of Squares } & \text { df } & \text { Mean Square } & \text { F-ratio } \ \text { Regression } & 8.34647 & 1 & 8.34647 & 0.045 \ \text { Residual } & 9817.18 & 53 & 185.230 & \ & & & & \ \text { Variable } & \text { Coefficient } & \text { SE(Coeff) } & \text { t-ratio } & \text { P-value } \ \text { Intercept } & 24.0487 & 11.40 & 2.11 & 0.0396 \ \text { Long } & 0.026186 & 0.1234 & 0.212 & 0.8327 \end{array}$ 
Test the null hypothesis that the true coefficient of Long is zero in this regression. State the
null and alternative hypotheses and indicate your procedure and conclusion.

Accepted Answer

The answer of Now, consider longitude. Should the longitude of...

Quiz 8: Regression, Associations, and Predictive Modeling

What is the predicted number of skiers for a Saturday with a temperature of 40° F. and a snow cover of 25 inches?

If the diameter is increased from 4 inches to 14 inches, the predicted number of cheerios will Increase by about…

Are the two sports teams mean ACT scores different?

Check the conditions for the regression and comment on whether or not they are satisfied.

Which of the explanatory variables appear to be associated with the number of skiers, and which do not? Explain how you reached your conclusion.

straight so it is reasonable to assume that the regression errors follow the Normal model. -If you think that the temperature might affect attendance differently on weekends than on weekdays, how would you change the regression to test this?

Compute a 95% confidence interval for the slope of the variable Weekend, and explain the meaning of the interval in the context of the problem.