University of Illinois at Chicago / College of Business Administration
MBA 503: Statistics / Fall, 1997 / Sclove
Textbook: Levine, Berenson & Stephan

Midterm Examination
9:30-11:30, Thursday, 18-September

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Try to sit only in every other row, every other seat.

Open book: You may use books and your notes. Work by yourself.

This exam is based on Chapters 1 through 7 of the textbook and the related notes, lectures and
homework.  

Five parts, 20 points per part.
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PART 1.  [20 pts.]  DISTRIBUTIONS FOR A LIKERT SCALE

1.  [3 pts.] Write down a distribution on a five-point Likert scale that shows a lot of consistency
of responses (no fair putting all the probability on one value!).

TABLE.  Distribution
     ------------------------------
value:  1     2     3     4     5

prob:  __    __    __    __    __
     ------------------------------

2.  [3 pts.] Compute the mean of this distribution.



3.  [3 pts.] Compute the variance of this distribution.



4.  [1 pt.]  Take the square root of the variance to obtain the standard deviation.


5.  [3 pts.]  Write down a distribution on a five-point Likert scale that shows a lack of
consistency of responses.

TABLE.  Distribution
     ------------------------------
value:  1     2     3     4     5

prob:  __    __    __    __    __
     ------------------------------

6.  [3 pts.]  Compute the mean of this distribution.


7.  [3 pts.]  Compute the variance of this distribution.



8.  [1 pt.]  Take the square root of the variance to obtain the standard deviation.


PART 2.  [20 pts.]  RANDOM SAMPLING FOR MEASURED CHARACTERISTICS; 
                         THE NORMAL DISTRIBUTION

     In quality control, samples are selected from a production  line and  various  quality 
characteristics are measured in order to check that the process is "in control." Suppose that a
bottling process  is intended  to  fill bottles with,  on average,  21  fluid ounces of beverage. 
Variation around this mean follows the normal distribution with a standard deviation of 0.5 fluid
ounces.

9.   [5 pts.]  What is the standard error of the mean if n = 25 ?





10. [15 pts.]   If a technician samples 25 bottles (when the process is "in  control")  and  measures
the amount of beverage in each, what is the probability that the sample average (for  the  25 
bottles)  will exceed 21.05 fluid ounces?







PART 3.  [20 pts.]  STRATIFIED SAMPLING CASE, CONTINUED

Suppose the N's and s's are as before, N1 = 1550, N2 = 620, N3 = 930 households, s1 = 5, s2 = 15,
s3 = 10 hours, but the costs are equal.  The optimal sample allocation is then called "Neyman
sampling," after the famous statistician Jerzy Neyman, one of the founding fathers of modern
statistics.

11.  [15 pts.]  Compute the optimal sample sizes if the total n is 100.














12.  [ 5 pts.]  In this optimal allocation,  n2  =  n3.   Why?








PART 4. [20 pts.]  INTERVAL ESTIMATION OF A PROPORTION

13.  [5 pts.]  Briefly,  what is the "margin of error"?






14.  [15 pts.]  In political polls regarding elections with two candidates the percentages are fairly
close to 50%.  In a typical such poll the sample size is taken to be 1600.  Why in such a poll is
the margin of error often stated to be about three percent?





PART 5.  [20 pts.]  ACCEPTANCE SAMPLING

Let  p  be the unknown true proportion of defectives in the lot.  Consider testing the null
hypothesis p = .03 against the alternative p = .07, based on a sample of n = 200  from a lot of N =
2000.  Construct the .05 level test, as follows.  Let  X  be the number of defectives in the sample.

15.  [ 8 pts.]  Under the null hypothesis,  what is the standard deviation of  X ?  HINT:  Don't
forget the finite population correction.  





16.  [2 pts.]  Under the null hypothesis, what is  the mean of X ?  HINT:  This is just  np, where 
n = 200 and  p = .03.  



17.  [10 pts.]  One accepts the null hypothesis if  X  is less than or equal to  c.  This number  c  is
called the "acceptance number."  What is the value of  c  here?  HINT:  X has a distribution that
is well approximated by a normal distribution with the mean and standard deviation you
computed above.  



















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12-Sept-1997
midterm.mba503
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