252solngr3-041 3/19/04 (Open this document in 'Page Layout' view!)Name:

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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 the following cases, identify and and identify which method to use. If the hypotheses involve a mean, state the hypotheses in terms of both and . If the hypotheses involve a proportion, state them in terms of both and . If the hypotheses involve standard deviations or variances, state them in terms of both and or . 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) - especially the new rules.

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Example: This may seem long but it appears on last year’s 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 is the mean of the second sample, you are hoping that , which, because it contains no equality is an alternate hypothesis. So your hypotheses are or . If , then . The important thing to notice here is that the data are in before and after pairs, so you use Method D4.

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1. You have data on income in two villages ( in village 1, 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. and you have to decide whether to use the mean or the median income.

Solution: If is the median. . Since we are comparing medians and the data are not paired, use Method D5a.

252solngr3-041 3/19/04

2. You have a sample of earned incomes for 25 couples, both of whom are teachers. ( is the women's incomes in a column, is the men's. Each line represents one couple. ) Test to see if the men make more than the women.

Solution: If is the median. . Since we are comparing medians and the data are paired, use Method D5b.

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. You want to show that businesses in the Northeast are more likely to export. ( is the total number of firms that export in the Northeast sample, in the Southeast).

Solution: If or . If , then . Since we are comparing proportions, use Method D6.

4. You expand the sample in 3 by adding 60 small businesses in the Midwest, ( is the number of these that export). You test the hypothesis that the same fraction of businesses export in each region.

Solution: If or . This is a chi-squared test of homogeneity. Since we are comparing multiple proportions, use a chi-squared test.

5. You have profit rates, , for a sample of 20 pharmaceutical firms in Europe and profit rates, , 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.

Solution: or . If , then . Because you believe that the Normal distribution applies, you use a method that compares means. The total sample size is too small to use Method D1, which means that D2 or D3 should work. You could test the variances for equality and use D2, or not bother and use D3.

6. In order to see which garage to use under contract for automobile repairs, 10 cars are towed first to garage 1 and than to garage 2. You end up with two data sets, the first data column, , is estimates from the first garage and the second data column,, 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.

Solution: There is no reason to assume that one garage is cheaper than the other, so or . If , then . Again, you compare means because you are, presumably, interested in the total amount that you will pay for the repairs, which means that you want the lowest average cost. The important thing to notice here is that the data are in pairs, so you use Method D4.

252solngr3-041 3/19/04

7. You are having a part produced in two different machines. is 200 randomly selected data points that represent the length of parts from machine one, 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. You could use Method D2 (if you tested the variances for equality) or D3 here, but, since you have two large samples, it would be far easier to use Method D1.

Solution: or . If , then .

8. You also suspect that parts from machine two are more variable in length than parts from machine one. (This is the same as saying that machine 2 is less reliable than machine 1). Test this suspicion.

Solution: or . In terms of the variance ratio or , the alternate hypothesis rules, so and . Since you are comparing variances, use Method D7.

Paired Samples / Independent Samples
Location - Normal distribution.
Compare means. / Method D4 / Methods D1- D3
Location - Distribution not Normal. Compare medians. / Method D5b / Method D5a
Proportions / Method D6
Variability - Normal distribution. Compare variances. / Method D7

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