Jaimie Kwon Page 10 10/28/2004
Stat {4601, 6872} Midterm (90 min.) Name____________
• Open book and open note; Use a simple calculator if necessary.
• Show your work. (E.g. Next to each numerical answer, you need to put things like P0.01(B≥3) or
something equivalent) An asterisks (*) means a hard question.
The following are head width measurements (in micrometer) of 10 common mayfly species,
Stenacron
interpunctatum
, in a certain habitat.
# Head Width # Head Width
"1" 36
"2" 31
"3" 30
"4" 27
"5" 20
"6" 33
"7" 27
"8" 18
"9" 19
"10" 28
#1. (60 points) Use procedures based on signed ranks to answer the following questions.
(a) What assumptions do you make about the distribution of the measurements when you employ these
procedures?
(b) Test the hypothesis that the median head width θ for mayflies from the habitat is 22 µm against
alternative that it is greater than 22. Compute the P-value.
(c) [
Use only the five numbers on the first column for the next 3 problems
] Obtain a point estimate
of θ.
(d) [
Use only the 5 numbers
] Find a lower confidence bound for θ with confidence coefficient .906.
(e) [
Use only the 5 numbers
] Obtain a confidence interval for θ with confidence coefficient .938.
(f) [
Use only the 5 numbers
] Use the large sample approximation to the confidence bound for θ with
the same confidence coefficient as in (e). Be sure to be conservative.
#2. (60+10 points) Use procedures based on signs to answer the following questions. (Use all 10 numbers )
(a) What assumptions do you make about the distribution of the measurements if you employ these
procedures?
(b) Test the hypothesis that θ is 22 µm against alternative that it is greater than 22.
(c) * What is the power of the test when the true θ is 25 µm?
(d) Obtain a point estimate of the θ.
(e) Find a lower confidence bound for θ with confidence coefficient .9453.
(f) Obtain a confidence interval for θ with confidence coefficient .9786.
(g) Use the large sample approximation to compute the confidence bound for θ with the same
confidence coefficient as in (f). Be sure to be conservative.
#3. (40+30 points) Use procedures based on Z-test (not t-test) to answer the following questions,
assuming population standard deviation is know to be σ=6.12. (Use table A.1) Use the pre-computed sample
mean
Z
=26.95.
(a) What assumptions do you make about the distribution of the measurements if you employ these
procedures?
(b) * Which of the three assumptions made so far (#1 (a), #2 (a), #3 (a)) is strongest? Which is
weakest? What’re the advantages of using procedures based on weak assumptions? What’re the
advantages of using ones based on strong assumptions?
(c) * How would you decide which of the three procedures (#1, #2, #3) to use?
(d) Test the hypothesis that θ is 22 µm against alternative that it is greater than 22.
(e) * What’s the power of the test when the true θ is 25 µm?
(f) Obtain a point estimate of θ.
(g) Obtain two confidence intervals for θ with confidence coefficient .938 and. 9786, respectively.
