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**Summary of assignment**

*Post in this topic ONE realistic and non-frivolous management example of just one type of hypothesis using the population mean: either 1 tail upper or lower. Your variable must be quantitative not qualitative so this means u not p which would be percentages and proportions. Only use quantitative data. You do not need to do any calculations or list any formulas. Justify why ?your testing? of this hypothesis would be worth testing so stay away from frivolous examples or population sizes of less than 500 (N>500). I recommend your pretend sample of be about 50(n=50). Include your sampling methodology and management decision you will make if you can accept Ha (see the paragraph above Item 1). The instructions below are probably longer than what you need to write and should most likely be less than one page. I will provide one example but only use it as a guide and don?t just copy my example and change a few words. I do not expect you to write a lot. Keep it as short as possible. I am interested in you showing me you understand the process and can communicate basic results to others.*

Please use this format below and label Parts 1 through 4. Use the following format and be sure to label the parts the same way I did so it can facilitate grading and discussion:

Part 1a

Part 1b

Part 1c

Part 1d

Etc

Part 2a

Part 2b

Etc

Same format for Parts 3 and 4. Be sure they match below.

**Assignment Instructions**

**PRIOR TO YOU PRETENDING TO DRAW A SAMPLE**:

*General Instructions for Parts 1a to 1g, *

*Preparation: Briefly explain the important problem/issue you would test for (make up numbers). At this time, you have not collected any data in your pretend scenario. What is the problem you want answered? Identify the QUANTITAVE variable you will collect in your sample to answer your question? You will most likely pretend to do a survey but only tell me about the one question that will give you data for the the variable. State what course of action your management team will make if Ho is rejected.*

Part 1a (problem, issue or some opportunity in a couple of complete sentences):

Part 1b: (Based on above what is the question you want answered and how does it tie into the decision you want to make?)

Part 1c: (What is your quantitative variable and unit (not percentages) you will measure and should be tied in to 1b?)

Part 1d: (What will be your course of action if Ho is rejected or not rejected. Use complete sentences).

Part 1f: (List your numeric hypotheses (Ho and Ha) and be sure to use the proper format and include the units in your your two hypotheses. Remember your variable is quantitative and therefore no percentages or proportions.)

Part 1g: (List your descriptive hypotheses (Ho and Ha). The reader should be able to understand the basics of what you are testing by reading the narrative/descriptive statements so be clear).

*General Instructions for Parts 2a to 2d]*

*Sampling: explain exactly what is the population of interest and the basic procedures you will take to select a sample from the population. Don't be vague, describe the mechanics of selecting the data and the sample size (n) and this can be done in a few sentences. You cannot be vague because the reader needs to have a clear understanding of your methodology so could theoretically rerun your hypothesis test. Before you pretend to take a sample state the rejection value for alpha at .05 or the rejection value for .1, .05 or what every alpha you will use based on the amount of risk you are willing to take. So the alpha criteria will have an associated z or t value (t for n 30 or less). Once the sample is drawn you will compare this z or t value associated with an alpha level (like .05) with your test statistic. If you are taking a survey only tell me the question you will be using to measure the variable of interest. Don't tell me about the other questions, just one! Also, try to have a simple random methodology...make it easy on yourself!*

Part 2a (What is your population of interest. It should be at least N=500 or no need to take a sample, n)?

Part 2b: How will you draw the sample to insure it is a random sample? Be specific enough so someone would know how to draw the same sample. A common problem students have is selecting a sample that is not random):

Part 2c (sample size): n=

Part 2d: What is your alpha level (.05, .01, etc) and the associated z or t or t value? Also explain why you used to or z (based on textbook and instructor).

Part 2d: Based on 2c above, in one sentence what will be your rejection rule you will use to evaluate the ?test statistic? you get from your actual data?)

**NOW PRETEND YOU HAVE DRAWN THE SAMPLE. **

**General Instructions for Parts 3a to 3e)**

After you?ve drawn the sample, your evaluation. Pretend you now have the sample; State the test statistic (make it up but be consistent) and compare to your alpha value you have for 2c and tell the reader if you reject or fail to reject Ho. Be sure to use the words, "fail to reject" or reject Ho and a one sentence of the management/research action you will take. You will also make up a p value and again show again why you would reject or fail to reject Ho. Most importantly, tell us your management course of action (COA).

Part 3a (What is your test statistic which should be a z or t value? You make this up for this assignment but should be consistent). You will have something like 3.45z or maybe 9.21t but make it look credible based on your specific scenario.

Part 3b (What is you made up p value? It should be consistent with your computed test statistic)

Part 3c (Write: Compare your alpha value to the test statistic value and tell the reader if you reject or fail to reject Ho. Do use this terminology such as test statistic.

Part 3d (Write: Compare your alpha level (i.e. .01, .05, etc not the actual z or t values) to the p value you computed (pretended) and tell the reader if you reject or fail to reject Ho. Do use this terminology such as p value.

Part 3e (Based on 3c and d what is your recommended course of action for management or researchers? Make this no more than a couple of sentences):

?Lady Tasting Tea?1

R. A. Fisher was one of the founding fathers of modern statistics. One of his early, and perhaps

the most famous, experiments was to test an English lady?s claim that she could tell whether milk

was poured before tea or not. Here?s an account of the seemingly trivial event that had the most

profound impact on the history of modern statistics, and hence arguably, modern quantitative

science (Box 1978).

Already, quite soon after he had come to Rothamstead, his presence had transformed

one commonplace tea time to an historic event. It happened one afternoon when he

drew a cup of tea from the urn and o?ered it to the lady beside him, Dr. B. Muriel

Bristol, an algologist. She declined it, stating that she preferred a cup into which the

milk had been poured ?rst. ?Nonsense,? returned Fisher, smiling, ?Surely it makes

no di?erence.? But she maintained, with emphasis, that of course it did. From just

behind, a voice suggested, ?Let?s test her.? It was William Roach who was not long

afterward to marry Miss Bristol. Immediately, they embarked on the preliminaries of

the experiment, Roach assisting with the cups and exulting that Miss Bristol divined

correctly more than enough of those cups into which tea had been poured ?rst to prove

her case.

Miss Bristol?s personal triumph was never recorded, and perhaps Fisher was not satis?ed at that moment with the extempore experimental procedure. One can be sure,

however, that even as he conceived and carried out the experiment beside the trestle table, and the onlookers, no doubt, took sides as to its outcome, he was thinking

through the questions it raised: How many cups should be used in the test? Should

they be paired? In what order should the cups be presented? What should be done

about chance variations in the temperature, sweetness, and so on? What conclusion

could be drawn from a perfect score or from one with one or more errors?

The real scienti?c signi?cance of this experiment is in these questions. These are, allowing incidental particulars, the questions one has to consider before designing an experiment. We will look at

these questions as pertaining to the ?lady tasting tea,? but you can imagine how these questions

should be adapted to di?erent situations.

? What should be done about chance variations in the temperature, sweetness, and so on?

Ideally, one would like to make all cups of tea identical except for the order of pouring milk

?rst or tea ?rst. But it is never possible to control all of the ways in which the cups of tea

can di?er from each other. If we cannot control these variations, then the best we can do?we

do mean the ?best?? is by randomization.

? How many cups should be used in the test? Should they be paired? In what order should the

cups be presented? The key idea here is that the number and ordering of the cups should

allow a subject ample opportunity to prove his or her abilities and keep a fraud from easily

succeeding at correctly discriminating the the order of pouring in all the cups of tea served.

? What conclusion could be drawn from a perfect score or from one with one or more errors? If

the lady is unable to discriminate between the di?erent orders of pouring, then by guessing

alone, it should be highly unlikely for that person to determine correctly which cups are

which for all of the cups tested. Similarly, if she indeed possesses some skill at di?erentiating

between the orders of pouring, then it may be unreasonable to require her to make no mistakes

so as to distinguish her ability from a pure guesser.

An actual scenario described by Fisher and told by many others as the ?lady tasting tea? experiment is as follows.

1 Adapted from Stat Labs: Mathematical statistics through applications by D. Nolan and T. Speed, SpringerVerlag, New York, 2000 1 ? For each cup, we record the order of actual pouring and what the lady says the order is. We

can summarize the result by a table like this: Lady says Tea ?rst

Milk ?rst Order of Actual Pouring

Tea ?rst

Milk ?rst

a

b

c

d

a+c

b+d a+b

c+d

n Here n is the total number of cups of tea made. The number of cups where tea is poured

?rst is a + c and the lady classi?es a + b of them as tea ?rst. Ideally, if she can taste the

di?erence, the counts b and c should be small. On the other hand, if she can?t really tell, we

would expect a and c to be about the same.

? Suppose now that to test the lady, 8 cups of tea are prepared, 4 tea ?rst, 4 milk ?rst, and

she is informed of the design (that there are 4 cups milk ?rst and 4 cups tea ?rst). Suppose

also that the cups are presented to her in random order. Her task then is to identify the 4

cups milk ?rst and 4 cups tea ?rst.

This design ?xes the row and column totals in the table above to be 4 each. That is,

a + b = a + c = c + d = b + d = 4.

With these constraints, when any one of a, b, c, d is speci?ed, the remaining three are uniquely

determined:

b = 4 ? a, c = 4 ? a, and d = a

In general, for this design, no matter how many cups (n) are served, the row total a + b will

equal a + c because the subject knows how many of the cups are ?tea ?rst? (or one kind as

supposed to the other). So once a is given, the other three counts are speci?ed.

? We can test the discriminating skill of the lady, if any, by randomizing the order of the cups

served. If we take the position that she has no discriminating skill, then the randomization of

the order makes the 4 cups chosen by her as tea ?rst equally likely to be any 4 of the 8 cups

served. There are 8 = 70 (in R, choose(8,4)?see also Devore pp. 71?73) possible ways to

4

classify 4 of the 8 cups as tea ?rst. If the subject has no ability to discriminate between two

preparations, then by the randomization, each of these 70 ways is equally likely. Only one of

70 ways leads to a completely correct classi?cation. So someone with no discriminating skill

has 1/70 chance of making no errors.

? It turns out that, if we assume that she has no discriminating skill, the number of correct

classi?cations of tea ?rst (?a? in the table) has ?hypergeometric? probability distribution

(see help(dhyper) in R or Devore pp. 128?129). There are 5 possibilities: 0, 1, 2, 3, 4 for a

and the corresponding probabilities (and R commands for computing the probabilities) are

tabulated below.

Number of correct calls

0

1

2

3

4 R command

dhyper(0,4,4,4)

dhyper(1,4,4,4)

dhyper(2,4,4,4)

dhyper(3,4,4,4)

dhyper(4,4,4,4) 2 Probability

1/70

16/70

36/70

16/70

1/70 ? With these probabilities, we can compute the p-value for the test of the hypothesis that the

lady cannot tell between the two preparations. Recall that the p-value is the probability of

observing a result as extreme or more extreme than the observed result assuming the null

hypothesis. If she makes all correct calls,the p-value is 1/70 and if she makes one error (3

correct calls) then the p-value is 1/70 + 16/70 ? 0.24.

The test described above is known as ?Fisher?s exact test.? References

Box, J. F. (1978). R. A. Fisher: The Life of a Scientist. John Wiley & Sons, Inc., New York. 3

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