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34. Refer to Exercise 3 where we are trying to link mortgage interest rate (x) to home sales (y).
The estimated regression equation there was ŷ = 90 − 4x.
1.
Show the 90% confidence interval estimate of the “population” intercept, α. Interpret
your interval.
2. Show the 90% confidence interval estimate of the “population” slope, β. Interpret your
interval.
37. Refer to Exercise 9 where we were trying to link customer waiting time (x) to customer satisfaction
(y). The boundaries for a 95% confidence interval estimate of the “population” slope here are −6.271
and 2.671. Could we reasonably believe that the “population” slope, β, is actually 0? 4.2? −2.5?
40. Refer to Exercise 3 where we are trying to link mortgage interest rate (x) to home sales (y). The
estimated regression equation there was ŷ = 90 − 4x. Can we reject the null hypothesis that the
population slope, β, is 0 at the 1% significance level? Explain the implications of your answer.
44. Refer to Exercise 9 where we are trying to link customer waiting time (x) to customer satisfaction (y).
The estimated regression equation turned out to be ŷ = 126 − 1.8x. Is the sample slope (b = −1.8)
significantly different from 0 at the 5% significance level? Explain the implications of your answer.
51. Refer to Exercise 4, where we are trying to link average hourly wage (x) to employee turnover (y).
The estimated regression equation turned out to be ŷ = 139.2 − 6.7x. Show the 95% confidence interval
estimate of the expected turnover rate for the population of all companies with an average hourly wage
of $15.
57. Refer to Exercise 4, where we are trying to link average hourly wage (x) to employee turnover (y).
The estimated regression equation turned out to be ŷ = 139.2 − 6.7x. Show the 95% prediction interval
estimate of the turnover rate for a particular company with an average hourly wage of $15.
81. Partial regression results from a sample of 12 observations are shown below. Fill in the
missing values indicated by ( )*. Can we use the sample results shown here to reject a β = 0 null
hypothesis at the 5% significance level? Explain
83. For each of the following cases, report whether the coefficient for the variable x is
significantly different from 0 at the 5% significance level and explain the implications of
your decision.
1.
n = 13
2.
n = 19
3.
n = 25
ECON 3100
PROBLEM SET 4
Textbook questions: Chapter 11: 34, 37, 40, 44, 51, 57, 81, 83
Statistical investigation:
Your Econ 3100 classmate Kim would like to buy a used Volkswagon Eurovan for an extended
road trip this summer. Her neighbor Tim has offered to sell her his Eurovan. Tim’s Eurovan is
has 75,000 miles on it. Kim and Tim agree that the van is in average condition. Tim has asked
her to come up with a fair price. Your classmate has never heard of “blue book value” and sets
out to estimate the value using her newly acquired knowledge of regression analysis. She checks
the Seattle Times and finds a long list of Eurovans for sale. She selects a random sample of 10 vans,
ranging in mileage from 59,000 to 129,000 miles. For each Eurovan, she enters the mileage (in 1000s)
and the offered sales price (in $1000s) into JMP. She runs a regression predicting asking price in
thousands based on mileage in thousands. She determines the relationship between mileage and
asking price and also runs some estimates for the price of a Eurovan with 75,000 miles on it.
The data for this example is found in the DATA folder on Canvas, entitled Eurovan.jmp.
1. What is the estimated regression equation?
2. To what extent is variation in the prices of the vans explained by differences in their mileage?
3. Give a 95% confidence interval for the coefficient on mileage.
4. Interpret the confidence interval calculated in question 3. Exactly what does it tell you?
5. Sketch a graph illustrating the estimated relationship between mileage and a van’s price.
6. Give a point estimate of the predicted price for a Eurovan in average condition with 75,000
miles on it.
7. Give a 95% confidence interval for the average price of Eurovans with 75,000 miles on them.
8. Give a 95% confidence interval (prediction interval) for the price of an individual Eurovan
with 75,000 miles on it.
1
9. Explain carefully why the confidence interval you gave for the individual van is wider that
that you gave for the average van.
10. Assuming that your classmate and Tim agree that his van is in average condition, what price
should she offer him? What is the price you would consider fair? Explain.
11. The sample contains a Eurovan with 81,718 thousand miles on it. Assuming that the price
given accurately reflects the condition of the car, do you think this van is likely to be in belowaverage, average, or above average condition, given its mileage? Explain your answer.
12. Does the residual plot give any suggestion that one of the assumptions of the regression
model may be violated? If not, why not? If so, which of the assumptions may be violated, and
what in the residual plot indicates this.
13. Conduct a t-test at a 0.05 significance level as to whether mileage has a statistically
significant relationship to the price of a van. Be sure to provide the hypotheses, test statistics, pvalue and conclusion. Summarize the test result in a non-technical sentence.
14. Now suppose a researcher wanted to provide evidence using a 0.05 significance level, that
the price of a van declines by more than $250 for each additional 1,000 on the odometer. What
would be the appropriate hypotheses? The test statistic? The p-value? The conclusion?
2
Everyday statistics
Please read the article “Scientific Method: Statistical Errors.” You can find it online at:
http://www.nature.com/news/scientific-method-statistical-errors-1.14700
1. What is the definition of “p-value” given in the article?
2. According to the article, how did the p-value come into widespread use in scientific research?
3. The article mentions a study of online dating. What were the statistically significant findings
of that study?
4. Again, referring to the study of online dating mentioned in the article, to what extent were the
findings reported substantially significant? That is, how large were the differences described
between couples who met online and couples who did not?
5. The article describes a phenomenon called “p-hacking,” whereby researchers manipulate their
studies in order to reduce the p-values and produce statistically significant findings. List some of
the ways that researchers might manipulate their results in order to produce statistically
significant findings (low p-values).
6. What are some ways that researchers might go about improving their research, that is, what
might they do beyond just using p-values to establish the validity of their findings?
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eurovan - JMP Pro
File Edit Tables Rows Cols Analyze Graph Tools View Window Help
o
-6X
eurovan
D
Age
Mileage
(1000s)
64
Price
Type (full
(1000) camper/MV)
30 FC
1
9
9
28 MV
2
3
4
82.345
59
27 FC
12
8
76.767
5
Year
2000
2000
1997
2001
2001
2002
1995
2000
2001
2000
8
6
7
14
7
8
9
69.218
88.275
108.077
110.608
81.718
129.591
24.9 MV
21.992 MV
19.988 MV
15 FC
13.995 MV
10.999 MV
8.888 MV
9
8
10
9
Columns (570)
Year
Age
Mileage (1000s)
Price (1000s)
IL Type (full camper/MV)
Rows
All rows
Selected
Excluded
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Labelled
10
0
0
0
0
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