252grass3-072 10/23/07 (Open this document in 'Page Layout' view!)
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Graded Assignment 3
Part 1: In your outline there are 6 methods to compare means or medians, methods D1, D2, D3, D4, D5a
and D5b. Methods D6a and D6b compare proportions and method D7 compares variances or standard
deviations. In 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. One of these problems is a chi-squared test.
Note: Look at 252thngs ( 252thngs) on the syllabus supplement part of the website before you start (and
before you take exams). ). Neatness and clarity of explanation are expected. Note that from now on
neatness means paper neatly trimmed on the left side if it has been torn, multiple pages stapled and
paper written on only one side.
----------------------------------------------------------------------------------------------------------------------------Example: This may seem long but it appears on an old graded assignment 3.
A group of supervisors are given the exams on management skills before and after taking a course in
management. Scores are as follows.
Supervisor
Before
After
1
63
78
2
93
92
3
84
91
4
72
80
5
65
69
6
72
85
7
91
99
8
84
82
9
71
81
10
80
87
11
68
93
If we assume that the distribution of results is Normal, what method should we use to answer the question
“Has the course improved the scores of the managers?”
Solution: You are comparing means before and after the course. You can get away with using means
because the parent distributions are Normal. If  2 is the mean of the second sample, you are hoping that
 2  1 , which, because it contains no equality is an alternate hypothesis. So your hypotheses are
 H 0 : 1   2
 H 0 : 1   2  0
H 0 : D  0
or 
. If D  1   2 , then 
. The important thing to notice

H
:



H
:




0
2
2
 1 1
 1 1
H 1 : D  0
here is that the data are in before and after pairs, so you use Method D4.
-------------------------------------------------------------------------------------------------------------------------------1. You have data on income in two villages ( x1 in village 1, x 2 in village 2). You want to test the
hypothesis that village 2 has higher earnings than village 1. You know that income has an extremely skewed
distribution. so you have to decide whether to use the mean or the median income.
2. You have a sample of earned incomes for 25 couples, both of whom are teachers. ( x1 is the women's
incomes in a column, x 2 is the men's. Each line represents one couple. ) Test to see if the men make more
than the women.
3. You have interviewed a sample of 80 small businesses in the Northeast and 75 small businesses in the
Southeast. Each business has indicated whether they sell in foreign markets. 60 firms in the Northeast and
50 in the Southeast export. You want to show that businesses in the Northeast are more likely to export. ( x1
is the total number of firms that export in the Northeast sample, x 2 in the Southeast).
4. You interview a sample of 57 Pennsylvania businesses in 2002 and reinterview the same sample in
2007. You ask them whether they export. Your data consists of two items for each firm: whether they
exported in 2002 and whether they exported in 2007. You want to show that the proportion exporting has
increased. Of the 57 firms 30 exported in both years and 10 did not export in the first year, but did so in the
second. 4 firms discontinued exports after 2002.
5. You expand the sample in 3 by adding 60 small businesses in the Midwest, ( x3 is the number of these
that export). You test the hypothesis that the same fraction of businesses export in each region.
6. You have profit rates, x1 , for a sample of 20 pharmaceutical firms in Europe and profit rates, x 2 , for a
sample of 17 pharmaceutical firms in the US. You believe that they are normally distributed and you wish
to see whether the European firms were more profitable than the American firms.
7. In order to see which garage to use under contract for automobile repairs, 35 cars are towed first to
garage 1 and than to garage 2. You end up with two data sets, the first data column, x1 , is estimates from
the first garage and the second data column, x 2 , is estimates for the second garage. Each of the 10 lines of
data refers to one car. You believe that the estimates are approximately normally distributed. Compare the
estimates in garage 1 and 2.
8. You are having a part produced in two different machines. x1 is 200 randomly selected data points that
represent the length of parts from machine one, x 2 is 200 randomly selected data points that represent the
length of parts from machine two. You want to test your suspicion that parts from machine 2 are longer than
parts from machine 1. In a problem of this type you would assume that the lengths are normally distributed.
9. You also suspect that parts from machine one and machine two differ in the variability if their length.
(This is the same as saying that machine 2 and machine 1 are not equally reliable.) Test this suspicion.
10. You are going to do the exercise in 8) again, but this time you have done a test like that in Exercise 9)
and not rejected your null hypothesis. However, you have only 30 lengths from each machine.
Part 2: Do problems 3 and 4, using a 95% confidence level. Find p-values.
2