STAT177 Homework sheet #9 Solutions
1)
Here is the
appropriate output:
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Regression
model: |
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Vacation
= 0.1878 + 0.0434Income |
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2)
Here is the
appropriate output:
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Percentage
of variation in Vacation explained by the model: 93.55% |
3)
Here is the appropriate
output:
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Ho: the
model is not significant |
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Ha: the
model is significant |
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Reject Ho
if test statistic > 5.987 |
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Test
statistic = 86.999 |
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P-value =
0 |
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Reject Ho |
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Conclude
the model is significant |
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4)
Note that we input 75
and not 75000 in the input box since the variable is coded in thousands. Here
is the appropriate output:
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Predicting
Vacation |
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Income =
75 |
Prediction: |
3.4428 |
We then multiply 3.4428 by 1000 to get 3442.80 which
rounds to 3400.
5)
Here is the
appropriate output:
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95%
confidence interval for Vacation |
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Lower
limit: |
3.1788 |
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Upper
limit: |
3.7067 |
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As with question 4, we multiply the limits by 1000 to
get 3178.8 and 3706.7 which round to 3200 and 3700 respectively.
6)
Here is the
appropriate output:
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95%
prediction interval for Vacation |
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Lower
limit: |
2.9296 |
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Upper
limit: |
3.956 |
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Once again, we multiply the limits by 1000 to get 2929.6
and 3956 which round to 2900 and 4000 respectively.
7)
Here is the output of
the model:
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Regression
model: |
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grade =
24.3923 + 3.8911study + 2.9052sleep - 3.8384gender - 2.6087income |
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Percentage
of variation in grade explained by the model: 70.16% |
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Adjusted
for the number of variables: 62.2% |
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Ho: the
model is not significant |
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Ha: the
model is significant |
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Reject Ho
if test statistic > 3.056 |
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Test
statistic = 8.816 |
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P-value =
0.001 |
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Reject Ho |
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Conclude
the model is significant |
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95%
Confidence Interval |
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Coefficient |
Lower limit |
Upper limit |
P-value |
VIF |
Comment |
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Intercept |
24.3923 |
-1.171 |
49.9556 |
0.0601 |
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study |
3.8911 |
2.338 |
5.4441 |
0.0001 |
1.085 |
Keep this
variable |
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sleep |
2.9052 |
0.2589 |
5.5515 |
0.0335 |
1.07 |
Keep this
variable |
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gender |
-3.8384 |
-13.5456 |
5.8688 |
0.4126 |
1.0481 |
Drop this
variable |
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income |
-2.6087 |
-12.8851 |
7.6676 |
0.5964 |
1.0797 |
Drop this
variable |
From the output we see the model is grade = 24.3923
+ 3.8911*study + 2.9052*sleep – 3.8384*gender – 2.6087*income
8)
The percentage of
variation in grade explained by the model is 70.16%.
9)
Ho: B1 = B2 = B3 = B4
= 0
Ha: not all coefficients equal 0
Reject Ho if test statistic > 3.056
Test statistic = 8.816. As well, p-value = 0.1%
Reject Ho and conclude the model is significant.
10) From the t test section, we see the p-value for
study is 0.01%, that of sleep is 3.35%, that of gender is 41.26% and that of
income is 59.64%. Since study and sleep are the only variables with p-values
less than 5%, these are the only variables that are significant.
11) The highest VIF is 1.085. Since all the VIF are less
than 10, there are no collinearity problems.
12) Here is the output of the second model:
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Regression
model: |
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grade =
20.0198 + 3.9576study + 3.0188sleep |
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Percentage
of variation in grade explained by the model: 68.02% |
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Adjusted
for the number of variables: 64.26% |
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Ho: the
model is not significant |
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Ha: the
model is significant |
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Reject Ho
if test statistic > 3.592 |
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Test
statistic = 18.078 |
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P-value =
0 |
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Reject Ho |
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Conclude
the model is significant |
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95%
Confidence Interval |
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Coefficient |
Lower limit |
Upper limit |
P-value |
VIF |
Comment |
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Intercept |
20.0198 |
-3.0083 |
43.048 |
0.0842 |
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study |
3.9576 |
2.507 |
5.4083 |
0 |
1.0217 |
Keep this
variable |
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sleep |
3.0188 |
0.5298 |
5.5079 |
0.0203 |
1.0217 |
Keep this
variable |
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Predicting
grade |
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study =
10 |
Prediction: |
80.728 |
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sleep = 7 |
95%
confidence interval for grade |
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Lower
limit: |
75.5609 |
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Upper
limit: |
85.8951 |
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95%
prediction interval for grade |
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Lower
limit: |
59.7745 |
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Upper
limit: |
101.6814 |
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From the output, we see the model is grade = 20.0198
+ 3.9576*study + 3.0188*sleep
13) From the output, the average grade based on 10 hours
of study and 7 hours of sleep is 80.728 which rounds to 81.
14) The output for the 95% confidence interval ranges
from 75. 5609 to 85.8951 which round to 76 and 86 respectively.
15) The output for the 95% prediction interval ranges
from 59.7745 to 101.6814. The lower limit rounds to 60. However, since you can’t
have a grade higher than 100, the upper limit of the prediction interval is
100.
16) We construct a table:
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Adj. r2 |
ANOVA p-value |
t-test – are all significant? (yes/no) |
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Model #1 |
62.2% |
0.001 |
No |
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Model #2 |
64.26% |
0 |
Yes |
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Best |
#2 |
#2 |
#2 |
We choose model #2 since:
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it has the highest
adjusted r2
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it has the lowest
ANOVA p-value
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all its variables are
significant
§
it has no collinearity
problems