## Math.unb.ca

Department of Mathematics & Statistics
TIME: 3 hours. You are permitted to use one 8 1/2 × 11 “cribsheet”, but no other notesor papers, in this examination.

1. In a study of human blood types in non-human primates, a sample (assume it was
random!) of 71 orangutans were tested and 14 were found to have blood type B.

(a) Construct a 90% confidence interval for the proportion of individuals with blood
(b) A biologist asserts that one quarter of all orangutans have blood type B. Should
(c) How many orangutans would you need to sample to determine the proportion of
the population that have blood type B to within ±0.05 with 90% confidence?
2. In a study on the maturation of egg cells in the frog Xenopus laevis, oocytes from
each of 13 female frogs were divided into two batches. One batch was exposed toprogesterone, the other was not. Each batch was then assayed for cyclic adenosinemonophosphate (“cAMP”), a substance which can mediate cellular response to hor-mones. The data were entered into a Minitab worksheet. Following there are twoanalyses of these data (parts of which have been obliterated). One of these analysesis correct; the other is incorrect.

MTB > #----------------------------------------------------------MTB > # Analysis number 1:MTB > let c3 = c1 - c2MTB > name c3 ’Con-Pro’MTB > ttest c3;SUBC> alte **.

MTB > #----------------------------------------------------------

MTB > #----------------------------------------------------------MTB > # Analysis number 2:MTB > twos c1 c2;SUBC> pool;SUBC> alte **.

95% C.I. for mu Control - mu Progest: ( *****, *****)T-Test mu Control = mu Progest (vs *****): T= ****
Both use Pooled StDev = 1.60MTB > #----------------------------------------------------------
(a) Which is the correct analysis? Give a brief reason.

(b) Using the correct analysis, (you will need to supply some of the obliterated
items) test an appropriate hypothesis to answer the question “Does exposure toprogesterone tend to decrease the amount of cAMP?”. Conduct this test atthe 0.05 significance level. Be sure to give a complete answer to this question,including stating your null and alternative hypotheses clearly in symbols, andexplaining briefly in words what you conclude from the test.

3. In a population of bacteria 40% of the individuals are E. Coli and 14 % are E. Hys-
tolitica. Suppose that 30% of the bacteria in the population have diameter greaterthan 240 microns but that only 20% of the E. Coli have diameter greater than 240microns1. A bacterium is drawn at random from this population. Let C be the eventthat the bacterium drawn is an E. Coli, and H be the event that it is an E. Hystolitica.

(a) Are the events C and H mutually exclusive? Explain briefly.

(b) Are the events C and H independent? Explain briefly.

(c) What is the probability that the bacterium drawn is neither an E. Coli nor an
(d) Suppose that the bacterium drawn has diameter greater than 240 microns. What
is the probability that it is an E. Coli?
1These percentages have no basis in scientific reality.

4. In an article in The Journal of the American Medical Association the relationship
between smoking habits and household income was investigated for males aged 15years and older in the Minhang district of China. The subjects of the study wereclassified as “Current smokers”, “Former smokers”, and “Never Smokers”, and werealso classified into 5 income groups (1: less than 5000 yuan, 2: 5000 to 6999 yuan,3: 7000 to 9999 yuan, 4: 10,000 to 14,999 yuan, and 5: 15,000 yuan or more). Thecounts were analyzed in Minitab; the results, parts of which have been obliterated,are given below:
Expected counts are printed below observed counts
(a) Supply the missing entries in the Minitab output, labelled (i), (ii), . . . , (vi).

(Note that other entries, which you are NOT requested to supply, have beenobliterated as well.)
(b) State the appropriate null and alternative hypotheses clearly.

(c) On the basis of the Minitab output (including the bits supplied by you) bracket
the p-value for the hypothesis being tested,
(d) Summarize briefly in words, with reference to the “standard” significance levels
(i.e. 0.10, 0.05, and 0.01) what you conclude from this analysis.

5. In a study on the effect of amphetamine on water consumption a pharmacologist
injected 40 rats with amphetamine, and another 40 rats with saline (to comprise acontrol group). She then observed the amount of water each rat consumed over a24 hour period, and recorded these amounts, expressed as ml of water per kg bodyweight. The data were entered into a Minitab worksheet. Following there are twoanalyses of these data (parts of which have been obliterated). One of these analysesis correct; the other is incorrect.

MTB > #----------------------------------------------------------MTB > # Analysis number 1:MTB > let c3 = c1 - c2MTB > name c3 ’Con-Amph’MTB > tint c3
MTB > #----------------------------------------------------------
MTB > #----------------------------------------------------------MTB > # Analysis number 2:MTB > twos c1 c2;SUBC> pool.

95% C.I. for mu Control - mu Amphet: (****,
T-Test mu Control = mu Amphet (vs not =): T= ****
Both use Pooled StDev = 18.1MTB > #----------------------------------------------------------
(a) Which is the correct analysis? Give a brief reason.

(b) Using the correct analysis, (you will need to supply some of the obliterated
items) find a 95% confidence interval for the population mean difference in waterconsumption between control rats and amphetamine injected rats.

(c) Suppose it is claimed that in a 24 hour period control rats drink 20 ml/kg
more water than amphetamine injected rats, on average. On the basis of yourconfidence interval do you think this claim is true?

6. In a clinical trial on the effectiveness of two drugs (ticrynafen and hydrochlorithiazide)
in the treatment of high blood pressure each of the two drugs was given at either ahigh or a low dosage level to separate groups of subjects for 6 weeks. The responsewas recorded as drop (baseline minus final value) in systolic blood pressure (mm Hg)for each subject. The data were analyzed in Minitab, producing the following output,parts of which have been obliterated:
MTB > retr ’f:drugs’MTB > aovo c1-c4
Individual 95% CIs For MeanBased on Pooled StDev
(a) State (clearly) the hypotheses being tested.

(b) Supply the missing numbers from the locations labelled (i), (ii), . . . (vi) in the
(c) Bracket the p-value of the test.

(d) State your conclusions about the test, briefly in words, with reference to the
three “standard” significance levels (0.10, 0.05, and 0.01).

7. In a study on the relation of calcium concentration in blood cells and blood pressure,
observations were made on these variables for 27 subjects. A regression model was usedto
nanomoles — nM). The model was fitted in Minitab with the following results:
+---------+---------+---------+---------+---------+------calcium
MTB > regr c2 1 c1;SUBC> pred 90;SUBC> pred 110.

The regression equation isbld.pres = - 13.3 + 1.02 calcium

(a) Is there evidence in the data that mean blood pressure would differ from 0 when
calcium concentration is 0? Explain briefly. Also comment briefly.

(b) Find a 95% confidence interval for the mean change in blood pressure when
calcium contration changes by 1 unit (i.e. by 1 nM per platelet).

(c) Find a 95% prediction interval for the blood pressure of an individual when that
individual’s calcium concentration is 110 nM per platelet.

(d) Suppose that you were told that the average blood pressure for an individual with
calcium concentration of 90 nM per platelet was equal to 90 mm Hg. Would youbelieve this assertion? Explain briefly.

(e) Suppose that you were told that an individual with calcium concentration of 90
nM per platelet was observed to have blood pressure equal to 90 mm Hg. Wouldyou believe this assertion? Explain briefly.

8. For each of 12 horses, a veterinary anatomist measured the density of nerve cells at two
intestinal sites, one in the midregion of the jejunum, and the other in the mesentericregion of the jejunum. The densities (average counts over 5 equal sections of tissue)were as follows:
It is believed that nerve density is greater the midregion than in the mesenteric region.

Use the sign test to test an appropriate hypothesis. State the hypothesis clearly.

Bracket the p-value of the test. State your conclusions clearly.

Source: http://www.math.unb.ca/~rolf/Courses/04f/2264/oldfin1.pdf

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