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Assignment 2
Instruction:
- Due date: 12:45PM Saturday 13/04/2019 in class. Late assignment will NOT be accepted. You can submit
your assignment early to the department general office (E21-3034) any time when that office is opened.
- Use A4-size paper to write your answers. There is no need to hand in a copy of the questions. You can use
recycled A4 papers (sheets that were printed on one side) and write your answers on the blank side. If you use
other paper size, 10 points will be deducted. If you don’t have A4 papers, just ask me and I will give you
some.
- Put your answers in order (Answer to part a goes before answer to part b and so on). 10 points are deducted if
answers are not put in order.
- Staple all your sheets together. If not, 10 points are deducted and you are responsible if loose sheets get lost.
- Use 5 percent significance level unless otherwise indicated.
- You should use a calculator and go through all the calculation in this question. This provides practice for the
final exam. I suggest putting all the important statistics (SS, y hat, residuals, etc.) in one place (the same table)
for easy reference. Keep at least 3 digits after the decimal point; some answers will be more obvious if you keep
more digits.
To help determine how many beers to stock the concession manager at Yankee Stadium wanted to know how the
temperature affected beer sales. Accordingly, she took a sample of 10 games and recorded the number of beers
sold and the temperature (in Fahrenheit) in the middle of the game.
Temperature Beers
80 20,533
68 1,439
78 13,829
79 21,286
87 30,985
74 17,187
86 30,240
92 37,596
77 9,610
84 28,742
a) Estimate the intercept and slope coefficients and write down the estimated equation. [10 points]
b) Interpret the coefficients. [5 points] Does the sign of the slope coefficient make sense? Explain [5 points]
c) Calculate the sample coefficient of correlation between the temperature and the number of beers sold. [5
points] Use it to test whether there is a linear relationship between these two variables.[5 points]
d) Calculate the coefficient of determination [5 points]. Interpret it [5 points].
e) Calculate the standard error of the estimate (or root mean square error, RMSE). Is it large or small compared
with the average value of the dependent variable? [5 points]
f) Test the overall usefulness of this regression model by F-test. [5 points]
g) Test whether the temperature can explain the number of beers sold by t-test. [5 points]
h) Compare the test statistics in parts (c), (f), and (g) and explain why they are the same or different [5 points]
i) Calculate the predicted number of beers sold for the temperatures in the sample (that is, y ). Compare i y
with i y . Explain why they are the same or different [5 points] (Ignore rounding errors in this comparison.)2
j) Calculate the residuals from the regression (that is, i e ) and sum of the residuals i e . Do you think the value
of
i e make sense? Explain [5 points.] (Ignore rounding errors in this comparison.)
k) Make an estimation interval for the expected number of beers sold given a temperature of 100 degrees [5
points].
l) Make a prediction interval for the number of beers sold given a temperature of 100 degrees [5 points].
m) Which one, part (k) or part (l), is more reliable? Explain. [5 points.]
n) Would you prediction of the number of beers sold for a temperature of 110 degrees be more or less reliable
than the prediction in part (l)? Explain. [5 points]
o) Provide 95% confidence interval estimate for the slope coefficient of the regression.[5 points]
p) Calculate standard error of the estimate (or RMSE) from the residuals obtained through this formula ∑.
Compare the value you get here with that in part (e).
Assignment 2
Instruction:
- Due date: 12:45PM Saturday 13/04/2019 in class. Late assignment will NOT be accepted. You can submit
your assignment early to the department general office (E21-3034) any time when that office is opened.
- Use A4-size paper to write your answers. There is no need to hand in a copy of the questions. You can use
recycled A4 papers (sheets that were printed on one side) and write your answers on the blank side. If you use
other paper size, 10 points will be deducted. If you don’t have A4 papers, just ask me and I will give you
some.
- Put your answers in order (Answer to part a goes before answer to part b and so on). 10 points are deducted if
answers are not put in order.
- Staple all your sheets together. If not, 10 points are deducted and you are responsible if loose sheets get lost.
- Use 5 percent significance level unless otherwise indicated.
- You should use a calculator and go through all the calculation in this question. This provides practice for the
final exam. I suggest putting all the important statistics (SS, y hat, residuals, etc.) in one place (the same table)
for easy reference. Keep at least 3 digits after the decimal point; some answers will be more obvious if you keep
more digits.
To help determine how many beers to stock the concession manager at Yankee Stadium wanted to know how the
temperature affected beer sales. Accordingly, she took a sample of 10 games and recorded the number of beers
sold and the temperature (in Fahrenheit) in the middle of the game.
Temperature Beers
80 20,533
68 1,439
78 13,829
79 21,286
87 30,985
74 17,187
86 30,240
92 37,596
77 9,610
84 28,742
a) Estimate the intercept and slope coefficients and write down the estimated equation. [10 points]
b) Interpret the coefficients. [5 points] Does the sign of the slope coefficient make sense? Explain [5 points]
c) Calculate the sample coefficient of correlation between the temperature and the number of beers sold. [5
points] Use it to test whether there is a linear relationship between these two variables.[5 points]
d) Calculate the coefficient of determination [5 points]. Interpret it [5 points].
e) Calculate the standard error of the estimate (or root mean square error, RMSE). Is it large or small compared
with the average value of the dependent variable? [5 points]
f) Test the overall usefulness of this regression model by F-test. [5 points]
g) Test whether the temperature can explain the number of beers sold by t-test. [5 points]
h) Compare the test statistics in parts (c), (f), and (g) and explain why they are the same or different [5 points]
i) Calculate the predicted number of beers sold for the temperatures in the sample (that is, y ). Compare i y
with i y . Explain why they are the same or different [5 points] (Ignore rounding errors in this comparison.)2
j) Calculate the residuals from the regression (that is, i e ) and sum of the residuals i e . Do you think the value
of
i e make sense? Explain [5 points.] (Ignore rounding errors in this comparison.)
k) Make an estimation interval for the expected number of beers sold given a temperature of 100 degrees [5
points].
l) Make a prediction interval for the number of beers sold given a temperature of 100 degrees [5 points].
m) Which one, part (k) or part (l), is more reliable? Explain. [5 points.]
n) Would you prediction of the number of beers sold for a temperature of 110 degrees be more or less reliable
than the prediction in part (l)? Explain. [5 points]
o) Provide 95% confidence interval estimate for the slope coefficient of the regression.[5 points]
p) Calculate standard error of the estimate (or RMSE) from the residuals obtained through this formula ∑.
Compare the value you get here with that in part (e).