EXAM 3 – FORM A
STAT 211

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Possible critical values that may be needed are z 0.10 =1.28, z 0.05 =1.645, z 0.025 =1.96, t 0.10; 24 =1.318,
t 0.05; 24 =1.711, t 0.025; 24 =2.064,  02.975;24 =12.401,  02.95;24 =13.848,  02.90;24 =15.659,  02.025;24 =39.364,
 02.05;24 =36.415,  02.10;24 =33.196
The width of a casing for a door is normally distributed with a mean of 24 inches and a standard deviation of 1/8 inch. The
width of a door is normally distributed with a mean of 23 inches and a standard deviation of 1/16 inches. Assume that the
width of a casing door and the width of the door are independent. Answer the following 3 questions using this information.
1.
Which of the following is the expected value of the difference between the width of the casing and the width of the
door?
(a) 1
E(X-Y)=E(X)-E(Y)=24-23=1
(b) 22
(c) 23
(d) 24
(e) 25
2.
Which of the following is the standard deviation of the difference between the width of the casing and the width of
the door?
(a) 0.0117
(b) 0.0195
(c) 0.1083
2
(d) 0.1398
2
1  1 
Var ( X  Y )  Var ( X )  Var (Y )        0.1398
 8   16 
(e) 0.1791
3.
What is the probability that the difference between the width of the casing and the width of the door exceed 1/4 inch?
(a) 0
(b) 0.25
(c) 0.50
(d) 0.75
(e) 1


P(X-Y>0.25)= P Z 
0.25  1 
 =P(Z>-5.27)=1
0.1398 
Aircrew escape systems are powered by a solid propellant. The burning rate of this propellant is an important
characteristic. Specifications require that the mean burning rate must be µ=50 cm/s. We know that the standard deviation
of burning rate is =2 cm/s. The experimenter selects a random sample of n=25 from a normal population and obtains a
_
sample average burning rate of x =51.3 cm/s. Answer the following 4 questions using this information.
4.
What should be the sample size to obtain the 95% confidence interval for the true mean burning rate with the interval
width of 0.5?
(a) 8
(b) 16
(c) 62
 2z  
 2(1.96)2 
n=   / 2   
  245.86
 0.5 
 w 
2
(d) 246
2
(e) 266
5.
Which of the following is the 95% confidence interval for the true mean burning rate?
(a) (50.47 , 52.13)
EXAM 3 – FORM A
STAT 211
(b) (50.52 , 52.08) = 51.3  1.96
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2
25
(c) (50.56 , 52.05)
(d) (50.60 , 52.01)
(e) (50.65 , 51.96)
6.
Are the specifications for the true mean satisfied when you look at the 99% interval for the true mean burning rate,
(50.27 , 52.33)?
(a) Yes
(b) No
=50 does not fall in between the interval limits
7.
When the experimenter selected the random sample of 25, he/she computed the standard deviation of burning rate,
s=1.9 cm/s. Do the data confirm the known standard deviation when you compute the 95% confidence interval for the
true standard deviation?
=2 falls in the 95% C.I for :
(a) Yes
 24(1.9) 2 24(1.9) 2

,
 39.364
12.401


 =(1.4836 , 2.6432)


(b) No
8.
Which of the following is the area between 13.848 and 39.364 on the chi-square curve when the sample size is 25?
(a) 0.025
(b) 0.05
(c) 0.925 = 1-0.025-0.05
39.364 (13.838) gives you the area on the right (left) as 0.025 (0.05).
(d) 0.95
(e) 0.975
The weight of a small candy is normally distributed with a mean of 0.1 ounce and a standard deviation of 0.01 ounce.
Suppose that random sample of 16 small candies are placed in a package. Answer the following 3 questions using this
information.
9.
What is the expected value of package weight?
(a) 0.01
(b) 0.1
(c) 0.16

16
X

(d) 1.6 = E 
1
i

   E ( X i )  16(0.1)

(e) 16
10. What is the variance of package weight?
(a) 0.0016 =
(b)
(c)
(d)
(e)
 16

Var  X i   Var ( X i )  16(0.01) 2
 1

0.016
0.16
1.6
16
11. What is the probability that the average candy weight in the package is less than 0.1 ounce?
(a) 0
(b) 0.1
(c) 0.4
(d) 0.5 =
(e) 0.8

0.1  0.1 
_

P x  0.1  P Z 
  P( Z  0)
0.01 / 16 



EXAM 3 – FORM A
STAT 211
FALL03
An industrial engineer’s assistant made 50 random observations of the upholstery installation team in an automobile
assembly plant. During 12 of the observations the workers were arranging materials beside their workstation. Answer the
following 3 questions using this information.
12. Are the large sample conditions satisfied to construct the confidence interval for the true proportion of time installers
spends arranging materials?
(a) Yes
50(12/50)=12  10 and 50(38/50)=38  10
(b) No
13. Which of the following is the point estimate for the true proportion of time installers spend arranging materials?
(a) 1/5
(b) 1/12
(c) 1/50
(d) 5/50
(e) 12/50
14. What sample size is required for the width of a 95% confidence interval for the true proportion of time installers spends
arranging materials to be at most 0.10?
(a) 21
(b) 51
(c) 71
(d) 141
^
(e) 281
n=
^
4 z 2 / 2 p (1  p )
w2

4(1.96) 2 (0.24)(0.76)
=280.28
0.10 2
The U.S. Census Bureau produces estimates of total resident population for each state on an annual basis. The following is
the descriptive statistics obtained by using MINITAB software.
Variable
X: # of Births
Y: # of Deaths
n
51
51
Variable
X: # of Births
Y: # of Deaths
Minimum
6035
3142
Mean
79366
47958
Median
54318
34066
Maximum
529610
234012
Q1
18942
12831
TrMean
64220
41493
StDev
93998
48872
SE Mean
13162
6843
Q3
85356
57544
Answer the following 5 questions using this information.
15. Which of the following is the point estimate for the true median of deaths?
(a) 34066 = sample median
(b) 47958
(c) 54318
(d) 79366
(e) 234012
16. Which of the following is the MLE for the true standard deviation of deaths if the death data are normally distributed?
(a) 48390.49 =
(b)
(c)
(d)
(e)
50(48872) 2
51
48872
93071.89
93998
8662376475
17. Assume deaths data are normally distributed. Is the MLE for the true variance of deaths its unbiased estimator?
(a) Yes
EXAM 3 – FORM A
STAT 211
(b) No
FALL03
 (n  1) s 2  n  1
n 1 2
 
E
E(s 2 ) 
 2
n
n
n


18. The 95% and the 99% confidence intervals for the true average number of births are computed below. Which of the
following is the 99% confidence interval for the true average number of births?
(a) (45472.94 , 113259.06) 99% is wider than 95%
(b) (53567.79 , 105164.21)
19. Which of the following is the estimated standard error for the difference between the average number of births and the
average number of deaths?
^
^
(a) 14835.104
^
^
 X2  Y2
Var ( X ) Var (Y )
93998 2 48872 2
= Var ( X  Y ) 





n
m
n
n
51
51
_
_
(b) 11243.431
(c) 80294.157
(d) 105943.836
20. If the critical value is
z / 2 =0.72 in the two-sided confidence interval for the population mean, which of the following
is the corresponding confidence level?
(a) 0.2358
(b) 0.5284 = P(Z<0.72)-P(Z<-0.72)=0.7642-0.2358
(c) 0.7642
(d) 0.8446