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EXAM 3 – FORM A
STAT 211
SPRING2004
A machine produces metal rods used in an automobile suspension system. A random sample of 15 rods is selected, and the
diameter is measured. The resulting data are shown below.
8.24
8.23
8.20
8.21
8.20
8.28
8.23
8.26
8.24
8.25
8.19
8.25
8.26
8.23
8.24
The following is the descriptive statistics obtained using the MINITAB software.
Variable
diameter
n
15
Mean
8.2340
Median
8.2400
TrMean
8.2338
Variable
diameter
Minimum
8.1900
Maximum
8.2800
Q1
8.2100
Q3
8.2500
StDev
0.0253
SE Mean
0.0065
Answer questions 1 to 10 using this information.
1.
Which of the following is the point estimate of the true mean rod diameter?
(a) 8.1900
(b) 8.2338
(c) 8.2340 the sample mean
(d) 8.2400
(e) 8.2800
2.
Which of the following is the point estimate of the true median rod diameter?
(a) 8.1900
(b) 8.2338
(c) 8.2340
(d) 8.2400 the sample median
(e) 8.2800
3.
Which of the following is the point estimate of the true proportion of rods with diameter larger than 8.20?
(a) 0.2
(b) 0.4
(c) 0.6
(d) 0.8 =the sample proportion where there are 12 data values larger than 8.20 out of 15
(e) 1
4.
Which of the following is the estimated standard error of the average diameter of the 15 rods?
(a) 0.00064
(b) 0.0017
^
^
_
(c) 0.0065
Var ( x)  Var ( x) / n  0.0253 / 15
=
(d) 0.0253
5.
Are the large sample conditions to calculate the confidence interval for the true proportion of rods with diameter larger
than 8.20 satisfied?
(a) Yes
^
(b) No
n
^
p >10 but n (1  p ) <10
The normality test done using MINITAB software supports the normal distribution on this data. Use this extra information
to answer questions 6 to 10.
6. Which of the following is the MLE of the true variance of rod diameter?
(a) 0.000597 = (n-1)s2/n=14(0.0253) 2/15 for normally distributed data
(b) 0.00064
(c) 0.023613
(d) 0.0253
(e) 0.027537
EXAM 3 – FORM A
STAT 211
7.
SPRING2004
If the true standard deviation of rod diameter is known to be 0.025, how large a sample would have been required to
estimate the true mean rod diameter to within 0.005 with 95% confidence?
(a) 10
(b) 95
z  
 1.96(0.025) 
(c) 97 =   / 2   

 0.005 
 B 
2
2
(d) 99
(e) 195
8.
Which of the following is the 95% confidence interval on true mean rod diameter?
_
(a) (8.2200 , 8.2480) =
x  t / 2;n 1
s
n
 8.234  2.145
0.0253
15
(b) (8,2212 , 8.2468)
9.
If the sample was 55 rods instead of 15 rods with the same mean and the standard deviation, would the critical value
used to calculate 95% confidence interval of the true mean rod diameter be different?
(a) Yes. it would be a large sample and z critical value would be used
(b) No
10. I claim that the true standard deviation in rod diameter is 0.03. The 95% confidence interval of the true variance is
computed as (0.0003 , 0.0016). Do the data support my claim?
(a) Yes because the variance 0.032=0.0009 falls between the limits
(b) No




^


^




^




^




^




^


Let E   1   3 , E   2    / 2 , E   3     2 , Var  1   18 , Var 2   1 and Var 3   16 . Use this
information to answer the following 2 questions.
11. Which of the following is one of the unbiased estimators of  ?
^
(a)
 2 /2
^
(b)  3 +2 because the unbiased estimators are
^
^
^
 1 /3, 2  2
and
3
+2
^
(c) 3  1
^
(d)
2
(e)
3
^
12. Which of the following is the minimum variance unbiased estimator of  ?
^
(a)
 1 /3
(b)
2
^
^
(c) 2  2
^
(d)
3
(e)
 3 +2
^
^
^
^
because the unbiased estimators are Var(  1 /3)=18/9=2, Var(2  2 )=4(1)=4 and Var(  3 +2)=16
EXAM 3 – FORM A
STAT 211
SPRING2004
13. Which of the following is the 95th percentile of chi-square values for the sample size 17?
(a) 7.962
(b) 8.682
(c) 26.296 has the area below 0.95 and area above 0.05 with the df=16
(d) 27.587
(e) 28.845
14. What sample size would be required for the width of a 95% large sample confidence interval of the true proportion to
^
be at most 0.05, irrespective of
(a) 385
(b) 693
^
p?
^
4 z 2 p(1  p) 4(1.96) 2 (0.5)(0.5)
=  /2 2
is the conservative sample size

w
0.005 2
(c) 1537
(d) 2759
15. If you are trying to compute the 95% confidence interval of the true variance for normally distributed data, which of
the following critical values should not be used?
(a) z / 2
(b)
t / 2:n 1
(c)
 2 / 2:n 1
(d)  1 / 2:n 1
2
(e) Exactly two of the above
16. If the critical value is
(a) and (b) cannot be used
t / 2:n 1 =2.821 where n=10 in the two-sided confidence interval for the population mean, which
of the following is the corresponding confidence level?
(a) 0.01
(b) 0.02
(c) 0.98 the area of 2.821 and the area below -2.821 are both 0.01 then the central area is 1-0.02=0.98
(d) 0.99
(e) 0.995
17. Let X1,X2,…,Xn be a random sample from a continuous distribution with pdf, f ( x ) 
following is the MLE of ?
(a)
 n

1 /  X i 
 i 1 
n
(b)
X
i 1
i
_
 
(c) 1 /  x 
_
(d) x
_
_



(e) x/  x  1
L=Likelihood=
1
n

e
n
 xi / 
i 1
1

e  x /  . Which of the
EXAM 3 – FORM A
STAT 211
SPRING2004
n
Ln(L)=  n ln(  ) 
 xi
i 1

n
 xi
_
d ln( L)
n
   i 1 2  0 then solve for   x
d
 
18. Let X1,X2,…,Xn be a random sample from a continuous distribution with the mean, E(X)=1/, which of the following
is the MME for ?
(a)
 n

1 /  X i 
 i 1 
n
(b)
X
i 1
i
_
 
_
(c) 1 /  x 
where
1/= x
_
according to the method of MME then =1/ x
_
(d) x
_
_



(e) x/  x  1
19. If you like to compare the significant differences between the 2 nd exam averages (2) and the 3rd exam averages (3),
you would set =2 -3. If the 95% large sample confidence interval for  is computed as (-5,10), Are there
significant differences between those two tests according to the confidence interval?
(a) Yes
(b) No
because no differences if =0 which falls in between the limits of the confidence interval
20. The same sample data is used to calculate 99%, 90% and 95% confidence interval of the true mean. Which of those
intervals has the smallest bound on the error estimation?
(a) 90% narrower interval has smaller bound on the error estimation
(b) 95%
(c) 99%
(d) we do not have enough information to answer this question
21. Which of the following is incorrect for the confidence interval of the true variance?
(a) If the 90% confidence interval captures the true variance, 95% interval will definitely capture the true variance
(b) If the 99% confidence interval captures the true variance, 95% interval may not capture the true variance
(c) If the 90% confidence interval for the true mean captures the true variance, 95% interval will definitely
capture the true mean
capturing the true variance does not imply capturing the mean
22. Which of the following statements is incorrect?
(a) The bound on the error estimation is always half width
(b) To construct the general large sample confidence interval for the parameter, we need to have its estimator to be at
least approximately normally distributed, unbiased with the known standard error
(c) We compute the confidence intervals for the parameters because it has the precision and the reliability of estimation
(d) There is only one unbiased estimator for each parameter
23. Which of the following is the P(  <2.088 or
(a) 0.01
(b) 0.06
=0.01+0.05
(c) 0.94
2
 2 >16.919) where  2
is a chi-squared random variable with n=10?
EXAM 3 – FORM A
STAT 211
SPRING2004
(d) 0.95
(e) 0.99
2
_
24. Let X1,X2,…,Xn be a random sample from a continuous distribution with the mean  and variance  . If  x  is a
 
2
biased estimator for 2, which of the following is the amount of bias?
(a) 
(b) 2
(c) 2
(d) 2/n
(e) +2
_2
_2
2
2
 _    _ 
2
2




E  x   Var x    E  x  
  where bias is E  x    
n
n
    
 
 
2
25. Let X1,X2,…,Xn be a random sample from a normal distribution with the mean  and variance 2. Which of the
following is the MLE of 2 +2 ?
_
(a) x s
_
 
2
2
(b)  x   s
2
2
s2
_
x

(c)  
n
 
2
(n  1) s
_
(d)  x  
n
 
2
_
where the MLE for  is x and the MLE for 2 is
(n  1) s 2
n