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```Chapter 12 Assessment Answer Key
[1202]
TABLE 12-8
It is believed that GPA (grade point average, based on a four point scale) should have a positive linear
relationship with ACT scores. Given below is the Excel output from regressing GPA on ACT scores using a
data set of 8 randomly chosen students from a Big-Ten university:
Referring to Table 12-8, what is the predicted average value of GPA when ACT = 20?
The table provides the coefficients for the underlying regression equation fit to the data:
For a value of X = 20, the corresponding Y estimate is: 2.61
[1203]
What do we mean when we say that a simple linear regression model is “statistically” useful?
The linear model, where we try to predict the valiue of Y based on some X and coefficients is a
better predictor of Y than the sample mean, Ybar.
Chapter 12 Assessment
Page 1
Chapter 12 Assessment Answer Key
[1204]
TABLE 12-2
A candy bar manufacturer is interested in trying to estimate how sales are influenced by the price of
their product. To do this, the company randomly chooses 6 small cities and offers the candy bar at
different prices. Using candy bar sales as the dependent variable, the company will conduct a simple
linear regression on the data below:
Referring to Table 12-2, what is the estimated slope parameter for the candy bar price and sales data?
Equations 12-3 and 12-4 can provide the values. Using PhStat, linear Regression. With the ANOVA /
coefficients
1204
Regression Statistics
Multiple R
0.88540441
R Square
0.783940969
Square
Standard Error
0.729926212
16.29860664
Observations
6
ANOVA
df
SS
MS
Regression
1
3855.421687
3855.421687
Residual
4
1062.578313
265.6445783
Total
5
4918
Coefficients
Standard Error
t Stat
F
Significance F
14.51345897
0.018945781
P-value
Lower 95%
Upper 95%
Intercept
161.3855422
26.16068642
6.169010231
0.003505887
88.75183241
234.0192519
Price (\$)
-48.19277108
12.65017211
-3.809653392
0.018945781
-83.31527952
-13.07026265
The slope coefficient is highlighted.
Chapter 12 Assessment
Page 2
Chapter 12 Assessment Answer Key
[1205]
TABLE 12-11
A company that has the distribution rights to home video sales of previously released movies would like
to use the box office gross (in millions of dollars) to estimate the number of units (in thousands of units)
that it can expect to sell. Following is the output from a simple linear regression along with the residual
plot and normal probability plot obtained from a data set of 30 different movie titles:
Chapter 12 Assessment
Page 3
Chapter 12 Assessment Answer Key
Referring to Table 12-11, what are, respectively, the lower and upper limits of the 95% confidence
interval estimate for the average change in video unit sales as a result of a one million dollars increase in
box office?
These values are provided in the table. – the lower and higher limits on the number of units sold, in
thousands of videos: 3.3 to 5.36 thousand. Be very careful about units, though, as the equation
requires gross box office in millions.
[1206]
The sample correlation coefficient between X and Y is 0.375. It has been found out that the p-value is
0.256 when testing H0: ρ= 0 against the one-sided alternative H1: ρ > 0 . To test H0: ρ = 0 against the
two-sided alternative H1: ρ ≠ 0 at a significance level of 0.2, the p-value is:
This problem does not require a calculation. Rather, it’s a definitional concern. When you are
evaluating against a statistic with a symmetric distribution, going from a one-tail test to a two tail test
with a given significance will double the p-value, making the outcome more likely to have occurred by
chance.
(0.256)(2)
Chapter 12 Assessment
Page 4
Chapter 12 Assessment Answer Key
[1207]
TABLE 12-11
A company that has the distribution rights to home video sales of previously released movies would like
to use the box office gross (in millions of dollars) to estimate the number of units (in thousands of units)
that it can expect to sell. Following is the output from a simple linear regression along with the residual
plot and normal probability plot obtained from a data set of 30 different movie titles:
Chapter 12 Assessment
Page 5
Chapter 12 Assessment Answer Key
Referring to Table 12-11, what is the value of the test statistic for testing whether there is a linear
relationship between box office gross and home video unit sales?
The coefficient in question is the Gross value – that’s what relates Gross to home unit sales. The test
statistic for that coefficient is the t-stat appearing next to it, with a value around 8.6, representing a very
high confidence that the relationship exists, and that the null hypothesis of a zero slope should be
dismissed.
[1208]
Testing for the existence of correlation is equivalent to
Testing for the existence of the slope β1. Correlation, by definition, is the existence of a non-zero
relationship between the independent and dependent variables, which is represented by β1. .
Chapter 12 Assessment
Page 6
Chapter 12 Assessment Answer Key
[1209]
TABLE 12-11
A company that has the distribution rights to home video sales of previously released movies would like
to use the box office gross (in millions of dollars) to estimate the number of units (in thousands of units)
that it can expect to sell. Following is the output from a simple linear regression along with the residual
plot and normal probability plot obtained from a data set of 30 different movie titles
Chapter 12 Assessment
Page 7
Chapter 12 Assessment Answer Key
"
True or False: Referring to Table 12-11, the normality of error assumption appears to have been violated.
False. The text explains in section 12.5 that we need to at the residuals, the difference between the
observed and the predicted data for each value of x. Look at the Normal Probability. The apparent
clustering of the observations near the zero point, gradual spreading symmetrically as the Z values (x
axis) become more extreme, and a roughly equivalent number of points in the extreme values of Z, all
give some sense that this data is normally distributed. More advanced statistics classes provide formal
mechanisms for judging this.
Chapter 12 Assessment
Page 8
Chapter 12 Assessment Answer Key
[1213]
TABLE 12-2
A candy bar manufacturer is interested in trying to estimate how sales are influenced by the price of
their product. To do this, the company randomly chooses 6 small cities and offers the candy bar at
different prices. Using candy bar sales as the dependent variable, the company will conduct a simple
linear regression on the data below:
Referring to Table 12-2, what is the standard error of the regression slope estimate, Sb1?
Using PhStat, linear Regression. With the ANOVA / coefficients
1204
Regression Statistics
Multiple R
0.88540441
R Square
0.783940969
Square
Standard Error
0.729926212
16.29860664
Observations
6
ANOVA
df
SS
MS
Regression
1
3855.421687
3855.421687
Residual
4
1062.578313
265.6445783
Total
5
4918
Coefficients
Standard Error
t Stat
F
Significance F
14.51345897
0.018945781
P-value
Lower 95%
Upper 95%
Intercept
161.3855422
26.16068642
6.169010231
0.003505887
88.75183241
234.0192519
Price (\$)
-48.19277108
12.65017211
-3.809653392
0.018945781
-83.31527952
-13.07026265
The slope standard error is highlighted.
Chapter 12 Assessment
Page 9
```
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