1. An experimenter sought to find a "natural" fly repellent. Her basic observation consisted of isolating 20 houseflies in a jar whose lid was either (1) untreated cloth (the control treatment) or (2) cloth treated with a solution made with black pepper. After a fixed time, the number of repelled flies (beyond a fixed distance from the lid) out of 20 was counted. This was repeated ten times for each group, and these counts averaged to produce a single Y value for the group under the treatment.

This basic experiment was performed independently with 14 randomly selected groups of flies, seven groups randomly assigned to each treatment, in completely random order. The data are shown below:

Average number of repelled flies (of 20): 14 groups.

Trt1 Trt2

untreated cloth black pepper

0.2 7.7

3.2 11.4

3.3 5.5

4.2 4.2

2.7 7.3

1.9 7.0

2.0 9.0

1. (10) Should these samples be considered to be independent of one another? Justify your answer in a sentence or two. It is an important decision, hence the point value.
2. (10) Let  ₁ be the long-run mean number of repelled flies for treatment 1 over many tested groups; similarly define  ₂ for treatment 2. Which of the methods discussed in class would you use for inference (confidence intervals or hypothesis tests) on  ₁ - ₂ ? Justify your answer (NOTE: no need for any printouts here; if you make a plot, tell me what plot you made, and tell me what you see in the plot. If you do a formal test, tell me what test you did and tell me how it turned out, including the P-value).
3. (10) Using the method you selected and justified in parts (a) and (b), obtain and interpret (in widely understandable language) a 95% confidence interval for  ₁ - ₂ .
4. (10) Is there strong evidence here that the black-pepper solution has a higher long-run average number of repelled flies than the untreated cloth? Test appropriate hypotheses using the method you selected and justified in parts (a) and (b). Give the hypotheses in terms of  ₁ and  ₂ , the test statistic, the P-value, and an interpretation in widely understandable language.

2. In the setting of problem 1, suppose a larger experiment is being planned.

1. (10) How many groups n _T would you recommend to be used in order to obtain a 95% MOE of at most 1 fly using the pooled-variance t method with equal sample sizes?
2. (10) Repeat the above calculation using the F-distribution 80-percent upper bound for MOE.

1. A stratified random sample taken from a finite population of 180 payments produced the tabular summary shown below.

1. (5) Calculate the stratified-random-sampling estimate of the population mean .
2. (10) Calculate the (estimated) standard error of the estimate given in (a).
3. (5) Give an approximate 90% confidence interval for the population total.

4. Let \[{{\bar{Y}}_{1}}\] be the mean of a random sample of size n ₁ from a population with mean  ₁ and variance  ² . Similarly let \[{{\bar{Y}}_{2}}\] be the mean of a random sample of size n ₂ from a population with mean  ₂ and variance  ² . Finally let \[{{\bar{Y}}_{3}}\] be the mean of a random sample of size n ₃ from a population with mean  ₃ and variance  ² (so the populations have possibly different means but equal variances). We would like to estimate

 _  _  _  _ 

1. (10) Propose an estimator (a linear combination of \[{{\bar{Y}}_{1}}\] , \[{{\bar{Y}}_{2}}\] \[{{\bar{Y}}_{3}}\] ) and find its variance as a multiple of  ² .
2. (10) Suppose that we are planning to use your estimator from (a) in a study with n _T total observations to make inference on  through an =0.05 two-sided t-test of H ₀ : =0; the test will have n _T -3 degrees of freedom. Let n ₁ = n _T /6, n ₂ = 4n _T /6, n ₃ = n _T /6. If we feel confident that  = 4, what total sample size is required for 80% power for the test if in fact  = 2? Show your work

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