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Graded Assignment 3
In your outline there are 6 methods to compare means or medians, methods D1, D2, D3, D4, D5a and D5b.
Method D6 compares proportions and method D7 compares variances or standard deviations. In all the
following cases, identify H 0 and H 1 and identify which method to use. If the hypotheses involve a mean,
state the hypotheses in terms of both  and D  1   2 . If the hypotheses involve a proportion, state
them in terms of both p and p  p1  p 2 . If the hypotheses involve standard deviations or variances,
state them in terms of both  2 and
 12
 22
or
 22
 12
. All the questions involve means, medians, proportions or
variances. (Most problems are highly edited versions of problems in McClave, et. al.)
Note: Look at 252thngs (252thngs) in the syllabus supplement before you start (and before you take
exams).
1. We have the amount spent by a sample of 25 pharmaceutical firms on research in 2002 and again for the
same firms in 2004. We assume that the underlying distributions are Normal. Was the average amount spent
in 2004 above the average amount spent in 2002?
2. Assume that you had the same data and a similar task to Problem 2 but your preliminary analysis
indicated that the underlying distributions were highly skewed to the right.
3. You have the grades of a sample of 15 traditional university students and  2 grades of a sample of 13
non-traditional university students, do these differ on average? Assume that preliminary analysis indicates
that the variances of grades of the two groups are similar and presume an underlying Normal distribution.
4. Since the results of Problem 3 were inconclusive, you take new samples of 250 traditional university
students and 150 non-traditional students, again you assume that the underlying distribution is Normal, but
you do not bother to compare variances. You are still trying to find out if grades of the two groups differ.
5. A recheck of the data in Problem 4 indicates that the underlying distribution is far from Normal, but you
are still trying to find out if the grades of the two groups differ.
6. If 10 out of a group of 80 randomly chosen people who have received information on your product are
inclined to buy it and 12 out of a group of 81 randomly chosen people are inclined to buy your product after
receiving the same information and seeing your commercial, does the commercial increase the proportion
who will buy your product?
7. Have new procedures decreased the variability of delivery times? Two samples have been taken and you
know x1 and s1 taken before the new procedures were instituted, and x 2 and s 2 , taken afterwards. There
seems to be little difference between average delivery times before and after.
8. A paper company wishes to know whether a new procedure has decreased the amount of time it takes to
unload trucks. What are the hypotheses this implies? Two samples are taken with the results below. How do
we decide whether to use Method D2 or D3? What hypotheses, etc do we test to make this decision?
Old Method
New Method
n1  50
n 2  50
x1  25 .4 minutes
x 2  27 .3 minutes
s 2  3.7 minutes
s1  3.1 minutes
9. You have the following data and want to see if means are similar before and after. Assume that the parent
distributions are Normal.
Person
Before
After
1
82
92
2
60
72
3
55
57
4
97
104
5
79
89
10. An experiment at Duke University was conducted to see if ability to identify food by taste and smell
decreases by age. One food (mushed apple) was correctly identified by 81out of 100 students and by 51 out
of 100 older people.
Extra Credit: In Method D6, we assume that we are comparing proportions from two independent samples.
In the McNemar Test we compare two proportions taken from the same sample. Assume that two different
question 2
question 1
yes no
questions are asked of the same group with the following responses.
So, for
yes
 x11 x12 
x

no
 21 x 22 
example x 21 is the number of people who answered no to question 1 and yes to question 1.
x11  x12  x 21  x 22  n , p1 
z
x12  x 21
x12  x 21
H 0 : p1  p 2
x11  x12
x  x 21
and p 2  11
. If we wish to test
, let
H 1 : p1  p 2
n
n
but the test is valid only if x12  x 21  10
A famous example of this concerns a debate between candidates, question 1 is whether the respondent
supports candidate 1 before the debate and question 2 is whether the respondent supports candidate 1 after
question 2
question 1
yes no
the debate. The data is
and the question is whether the debate has changed the
yes
27 7
 13 28 
no


fraction supporting candidate 1. Write this out as a hypothesis test and do the test.
Note: This is a version of the Chi-Square Test – Recall that  2 

O  E 2
E
. If we take x11 and x 22 as
question 1
given, and assume that the null hypothesis is correct, then the table already given,
yes
no
question 2
yes no
 x11 x12 
x

 21 x 22 
is our O , and the numbers in the x12 and the x 21 slots must be equal for there to be no change in
2
question 1
preferences, so that out E is
yes
no
question 2
yes
no
x12  x 21  . This means that two of the four terms in

x11


2
x  x

21
 12
x 22 
2


2
2 


O  E 2
E
x12  x 21 2
x12  x 21
x  x 21 
x  x 21 


 x12  12

 x 21  12

2
2




2
are zero and the remaining terms are  

x12  x 21
x12  x 21
2
2
2
. But  2 has only one degree of freedom, and, since  2 is defined as a sum of z 2 , we can
take a square root and say z 
x12  x 21
x12  x 21
.
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