Download Stat 571 Midterm Exam I - October 2008 - Prof. Cecile M. Ane and more Exams Data Analysis & Statistical Methods in PDF only on Docsity!
Stat 571 First Midterm Exam October 7, 2008
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:
1 10
2 12
3 16
4 26
5 14
6 22
Total 100
- Twenty tulip petals were randomly chosen from a large tulip field in the Netherlands. The field was part of an experiment in breeding tulips to increase their redness. Each petal then underwent spectrophotographic analysis to determine its redness on a standardized scale of 0 to 100 redness units (where 0 = no red at all, and 100 = nothing but red, with values only on whole numbers). The original data were not available, but there exists the following summary information:∑ Let xi denote the redness value for petal i. then 20 i=1 xi^ = 1044, and^
i=1 x
2 i = 56172; and the following histogram (note that the ranges above each bar indicate the lower and upper endpoints of that bin): Histogram of Standardized Petal Redness
Redness
Frequency
30 40 50 60 70
0
1
2
3
4
5
[30−34]
[35−39] [40−44]
[45−49]
[50−54]
[55−59]
[60−64]
[65−69]
(a) Find the sample mean petal redness.
(b) Find the sample standard deviation of petal redness.
(c) How many petals in the sample had a redness of between 40 and 49 units, inclusive?
- Mark whether each item is True or False. If the item is False, briefly explain why.
(a) I have two events A and B, both on the sample space S. A ∪ B = S, so A and B must be mutually exclusive. True False
(b) Let X ∼ Bin(n, p). Additionally, let Y = 2∗X. It follows that Y ∼ Bin(2n, p). True False
(c) In a test of H 0 : μ = 10 vs. HA : μ 6 = 10, using X¯ as an estimator of μ, suppose we find ¯x = 15 and p-value = 0.0001. Even though this is a two-sided hypothesis test, there is more evidence that μ is greater than 10 than there is that μ is less than 10. True False
(d) Suppose we take a sample from a large population, and all the data values we get in our sample are negative. In this case, the sample variance as computed on our sample will be negative too. True False
(e) Suppose we take a sample of size n = 20 from a large population. A histogram with 25 equally sized bins will likely result in an appropriate graphical summary of this information, if the general shape of the distribution is of interest. True False
(f) We said in class that the normal approximation to the Binomial is appropriate when n ∗ p > 5 and n ∗ (1 − p) > 5, where n is the number of trials and p is the probability of success on each trial. The reason that n is required to be ‘less large’ here than in a typical application of the Central Limit Theorem (the book states n > 30), is because the normal approximation to the binomial is not based on the CLT. True False
- (a) For the following sampling scenarios, state whether the binomial distribution would describe the probability distribution of possible outcomes. If you answer no, briefly explain why. i. The number of red flowers in a square meter in a field. Yes No
ii. The number of red-eyed flies (which is the normal condition) among 150 Drosophila individuals drawn at random from a large population. Yes No
iii. The total number of red-eyed flies in five Drosophila families, each family consisting of 30 genetically related individuals, with the families chosen at random from a large population. Yes No
(b) Murphy’s law states that pieces of toast that are buttered on only one side have a higher chance of landing butter-side down when dropped. Let p be the probability of landing butter-side down. i. State a null and alternative hypothesis in terms of p that could be used to test Murphy’s law. Justify your choice of alternative.
ii. An experiment is finally conducted: n independent pieces of toast are buttered on one side and then dropped. Assuming p = .65 and the experimenter uses n = 11, calculate the probability that exactly 5 slices of toast land butter-side down.
iii. Suppose n = 98. Assuming p = .5, calculate the mean and standard deviation for the number of slices that land butter-side down in this experiment.
iv. Still assuming n = 98 and p = .5, calculate the probability that at least 61 slices land butter-side down.
- In European earwigs, the males sometimes have very long pincers protruding from the end of their abdomen.
(a) The graph below represents the distribution of pincer lengths in a particular population of European earwigs (in mm). The distribution is approximately normal. Each of the two gray areas represents 2.5% of this population, i.e. 2.5% of the area under the curve. Estimate the following quantities from the graph:
5 6 7 8 9 10
pincer length (mm)
probability density 5 6 7 8 9 10
i. The mean
ii. The standard deviation
(b) Based on your previous estimates, complete the following sentence: “In the population described in (a), about 70% of male earwigs have pincers of length at most mm.”
Show your work below.
(c) In a different population, n male earwigs have been sampled, measured, and the average of their n pincer lengths has been calculated. This procedure has been repeated many times and we have plotted below the frequency distribution of sample means based on three sample sizes: n = 1, n = 2 and n = 8. Identify which frequency distribution corresponds to each sample size. Explain the basis for your decisions.
(i) (ii) (iii)
Mean pincer length
frequency
3 4 5 6 7 8 9 10
0
10
20
30
40
50
60
70
Mean pincer length
3 4 5 6 7 8 9 10
0
20
40
60
80
100
Mean pincer length
3 4 5 6 7 8 9 10
0
10
20
30
40
50
60
