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Stat 571 Second Midterm Exam November 21, 2006
Name:
- The exam is open book and open notes.
- Do all your work in the spaces provided. If you need additional space for your work, indicate clearly where the additional work can be found.
- The parts within a problem are not necessarily sequential.
- To receive full credit, you must show your work.
- Do not dwell too long on any one question. Answer as many questions as you can.
For instructor’s use:
Total 100
- In queenless wasp colonies of Vesputa rufa, all wasps are workers and 20% of these workers lay eggs. In queenright colonies (colonies with a queen), the queen lays the eggs, but it is suspected that some workers may also lay eggs. In a queenright colony, 120 workers were sampled at random. Each wasp worker was dissected, and it was found that 11 workers had functional ovaries.
(a) Test the null hypothesis that 20% of all workers in this colony have functional ovaries versus the one-sided alternative hypothesis that fewer than 20% workers have functional ovaries. Indeed, it is believed that the presence of a queen can only decrease the number of fertile workers. Calculate the p-value and evaluate the evidence against the null hupothesis.
(b) Discuss any assumption that you made in 1a.
- A field experiment was conducted to compare the yields (bushels/acre) of two different varieties of wheat, A and B. The raw data are as follows:
Variety mean sd A 110.50 60.32 120.44 40.22 110.36 10.73 20.97 90.53 70.51 40. B 190.11 190.57 120.42 10.52 160.83 20.80 115.54 81.
(a) Below are 4 sets of normal-quantile plots, with A on the left, and B on the right. Which set is the correct set for this data, 1, 2, 3, or 4?
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150
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250
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(b) Based on the set you chose in (a), does the normality assumption seem to be valid in this case? Why or why not?
(c) Perform an independent two-sample test of the null hypothesis that the mean for group A is equal to the mean for group B, versus the two-sided alternative, at α = 0.05. If you believe that normality is okay, assume the variances are equal, and use a t-test. If you believe that normality is violated, use the Mann-Whitney test. In either case, say whether you would reject or not reject the null hypothesis.
- The philodendron is very hearty tropical plant, making it an excellent low-maintenance houseplant. A botanist is interested in whether the heartiness of the plant to low-water conditions is affected by the amount of sunlight it receives. Thirteen plants were randomly allocated to two treatment groups, with 5 in the low-light group, and 8 in the high-light group. The plants were completely denied water, and the number of days to visible signs of distress (brown or wilted leaves) was recorded. The data are as follows:
Treatment mean sd Low-Light 6 9 10 4 5 6.80 2. High-Light 3 7 13 10 8 6 11 8 8.25 3.
Perform Levene’s test at α = 0.05 to determine if the variances in the two groups can reasonably be assumed equal.
Over, please.
- A study is to be conducted to determine the impact of a generating station in Portage, Wisconsin, on air quality. SO 2 concentration is to be measured at several sites at similar distance from the power plant. After the data is collected, a test will be used a level α = 0.05 in order to test the null hypothesis that the mean SO 2 concentration in the air is equal to 10 μg SO 2 /m^3. This is thought to be an acceptable SO 2 concentration. The study aims at verifying that the power plant is still “clean” after several years of operation. A one-sided alternative (mean SO 2 concentration exceeds 10 μg SO 2 /m^3 ) will be used for the test since it is believed that the power plant can only increase air pollution.
(a) A value of 30 μg SO 2 /m^3 is thought to be uncceptable. It is desired that such a level of SO 2 pollution have a 90% chance of being detected by the experiment. How many sites need to be sampled in order to achieve this goal? Note: Previous data collected near other power plants have shown that the standard deviation in SO 2 concentration is 18 μg/m^3 (variance of 18^2 ).
(b) If the test is to be done at level α = 0.01, and if it is still desired that a level of 30 μg SO 2 /m^3 be detected with a 90% chance, should one sample fewer, the same number of or more sites than in 5a? Explain concisely.
(c) If the test is made at level α = 0.05 as in 5a, but if the number of sites sampled is four times larger than in 5a, which level of SO 2 pollution will have a 90% chance of being detected: 7.5, 15 or 20 μg SO 2 /m^3? Explain concisely. Hint: There is no need of more calculations than using the formula for the sample size.
