PRINT NAME _______________
STA 4322/5328 – Spr/2010 Mini Exam1
For all problems, the sample mean, sample variance, and sample proportion are:
Sample Mean : Y 
1 n
 Yi
n i 1
Sample variance :

1 n
 Yi  Y
n-1 i 1

^
2
Sample Proportion : p 
Y
where Y ~ Bin (n, p)
n
Q.1: A spectrometer gives readings that are normally distributed with mean equal to the true weight if the
object being measured, and a standard deviation of 0.5. Assume repeated measurements are independent.
P1.a. If a lab technician makes 4 readings of the spectrometer on a particular item, what is the sampling
distribution of the sample mean, the average of the 4 readings? Label the true weight as Y*.


0.52
2

Y ~ N  Y  Y *, Y 
 0.0625 
4


P1.b.What is the probability that the sample mean will lie within 0.2 of the true mean?


  0.2
Y Y *
0.2 
P  0.2  Y  Y *  0.2  P


 P 0.8  Z  0.8  20.5  PZ  0.8
 0.0625


0
.
0625
Y


 2(0.5  0.2119)  2(.2881)  .5762




P1.c. How many readings will she need to take if she wants the probability that the sample mean lies
within 0.1 of the true weight with probability=0.90?


  0.1
Y Y *
0.1 
0.90  P  0.1  Y  Y *  0.1  P


 P 1.645  Z  1.645
 0.5 n


0
.
5
n
Y


0.1

 1.645  n  1.645(5)  8.225  n  (8.225) 2  67.65  68
0.5 n




Q.2: A swimmer is interested in the variation in his time to complete the 100 meter freestyle. His actual times
are normally distributed with sec and 2 = 16 sec2. His times are independent across days.
P2.a.He has a friend time him on 8 days, and obtains the sample mean and variance. Give the values a1
and b1 such that: P(a1 ≤ S2 ≤ b1) = 0.95
(n  1) S 2
2
~  n21
 72 (.975)  1.690  72 (.025)  16.013


(n  1) S 2 7 S 2
16.013(16) 
 1.690(16)
2

0.95  P1.690 

 16.013   P
 S2 
  P 3.86  S  36.60
2
16
7
7







P2.b.He swims at 2 different pools and is wondering if his distribution of times differ at the 2 pools. He
has his friend measure him on 8 days at each pool (in random order to avoid possible long-term
improvement/decay confound). In fact, there are no pool effects: 
P2.b.i.What is the probability his sample means for the 2 pools will differ by more than 3
seconds in absolute value?


 3
X  Y  1   2  3 
 16 16

X  Y ~ N  0,   4   1  P  3  X  Y  3  1  P


 4
 X Y
4 
 8 8


 1  P 1.5  Z  1.5  2 PZ  1.5  2(.0668)  .1336




S2
P2.bii. Give values a2 and b2 such that: P a 2  12  b2   0.95
S2


2


S1
S12
1
~ F7,7 F7,7 (.025)  4.995 F7,7 (.025) 
 .200  P 0.200  2  4.995   .95
2
4.995
S2
S2


Q.3. A lot acceptance sampling plan for large lots specifies that 64 items be randomly selected and that the lot
be accepted if no more than 10 of the items selected do not conform to specifications. What is the approximate
probability that a lot will be accepted if the true proportion of non-conforming items in the lot is 0.20? Use the
continuity correction in your calculation.
Y ~ Bin (n  64, p  0.2)  N (   np  12.8,  2  np(1  p)  10.24)
10.5  12.8


PY  10 | Y ~ Bin (64,0.2)   PY  10.5 | Y ~ N (12.8,10.24)   P Z 
 0.72   .2358
10.24


Q.4. The reading on a voltage meter connected to a test circuit is uniformly distributed over the interval
( where  is the true but unknown voltage of the circuit. Suppose that Y1,…,Yn denote a random sample of
such readings. (Hint: Y~U(a,b)  E(Y) = (b+a)/2 E(Y2) = (b2+ab+a2)/3).
^
P.4.a. Show that   Y is a biased estimator of and compute the bias.
  (  1)


2  1
   0.5  E Y    0.5  B Y    0.5    0.5
2
2
^
P.4.b. Compute MSE  
 
E (Y ) 

2
V (Y )
^
 ^    ^ 
MSE    V     B  
 (0.5) 2
n
 
    

b 2  ab  a 2 b 2  2ab  a 2 1


4b 2  4ab  4a 2  3b 2  6ab  ab 2
3
4
12
2

b 2  2ab  a 2 b  a 
  (  1) 2 1
1  3n
 ^  1 12




 MSE   
 0.25 
12
12
12
12
n
12n
 
 
V (Y )  E Y 2  E (Y ) 
2
 
