Download Statistics 270 - SFU Statistics

Statistics 350
Sample Midterm 1
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Instructions:
1. Read all questions carefully
2. Define all variables/events used in your solutions
3. Show all of your work
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1. Consider the simple linear regression model Yi= β0 + β1Xi + εi (i=1,2,…,n),
where the εi’s are iid N(0,σ2). Suppose that β0=0.
a. What is the implication for the regression function?
b. How would the regression function plot on a graph?
c. Suppose that σ2 is known. State the likelihood function for a sample of
size n.
d. Suppose that σ2 is known. Derive the maximum likelihood estimator for
β1. (Note: show that it is a maximum)
2. Consider the simple linear regression model Yi= β0 + β1Xi + εi (i=1,2,…,n),
where the εi’s are iid N(0,σ2). Suppose that β1=0.
a. Derive the least squares estimator for β0.
b. Show that your estimate for part (a) is a minimum.
c. Is your estimate in part (a) an unbiased estimator? Prove or disprove that it is
unbiased for β0.
d. Calculate ∑ei for this model. What does this imply about the residuals and a
plot of residuals versus X?
3. A substance used in biological and medical research is shipped by air-freight to
users in cartons of 1,000 ampules. The data below, involving 10 shipments were
collected on the number of times the carton was transferred from one aircraft to
another over the shipment route (X) and the number of ampules found to be
broken upon arrival (Y). Assume the simple linear regression model is
appropriate.
X
1
0
2
0
3
1
0
1
2
0
Y
16
9
17
12
22
13
8
15
19
11
Some summary statistics for these data are:
A simple linear regression model was fit to these data. The computer output is
summarized below.
ANOVAb
Model
1
Regres sion
Error
Total
Sum of
Squares
160.000
17.600
177.600
df
1
8
9
Mean Square
160.000
2.200
F
72.727
Sig.
.000a
t
15.377
8.528
Sig.
.000
.000
a. Predic tors: (Constant), X
b. Dependent Variable: Y
Coeffi cientsa
Model
1
(Const ant)
X
Unstandardized
Coeffic ients
B
St d. Error
10.200
.663
4.000
.469
a. Dependent Variable: Y
St andardiz ed
Coeffic ients
Beta
.949
a. Test the hypothesis that there is a linear association between the number of
transfers for a shipment and the number of broken ampules. State the
hypotheses, value of your test statistic and your conclusion.
b. Estimate the expected number of broken ampules when there are 3 transfers.
c. Give a 95% confidence interval for the expected number of broken ampules when
there are 3 transfers for the shipment.
d. A customer for the shipping company claims that, on average, there are more than
20 broken ampules when there are 3 transfers for the shipments. Test this
hypothesis at a 5% significance level.
e. A plot of the residuals versus the number of transfers is shown below. From the
plot, is there any evidence that the assumptions of the simple linear regression
model are violated (explain your answer)?
2.00000
1.00000
Residuals
0.00000
-1.00000
-2.00000
-3.00000
0.00
0.50
1.00
1.50
X
2.00
2.50
3.00
4.