Biometry - STAT 305 ~ Assignment #4 (33 points)
1. Albino Rats (9 pts.)
Albino rats used to study the hormonal regulation of a metabolic pathway are injected with a drug that inhibits body synthesis of protein. Usually 4 out of 20 rats die from the
drug before the experiment is over. If 10 animals are treated with the drug find the following:
a) The probability that at least 8 will be alive at the end of the experiment. (2 pts.)
b) The probability that more than 4 die. (2 pts.)
c) The probability that at least one dies. (2 pts.)
d) If we used 100 rats in a study what is the mean and standard deviation of the number of rats that would die during the course of the experiment? Use the mean and standard deviation to give an interval that is very likely, say 95% chance, to cover the number rats that would actually die. (3 pts.)
2. Darwin’s Study of Cross- and Self-Fertilization (5 pts.)
These data are from Charles Darwin’s study of cross- and self-fertilization. Pairs of seedlings of the same age, one produced by cross-fertilization and other other by self-fertilization, were grown together so that members of each pair were reared under nearly identical conditions. The aim was to demonstrate the greater vigor of the cross-fertilized plants. The data given in pairs are the heights of each plant (in inches) after a fixed period of time.
1 / 23.5 / 17.4
2 / 12.0 / 20.4
3 / 21.0 / 20.0
4 / 22.0 / 20.0
5 / 19.1 / 18.4
6 / 21.5 / 18.6
7 / 22.1 / 18.6
8 / 20.4 / 15.3
9 / 18.3 / 16.5
10 / 21.6 / 18.0
11 / 23.3 / 16.3
12 / 21.0 / 18.0
13 / 22.1 / 12.8
14 / 23.0 / 15.5
15 / 12.0 / 18.0
Research Question: Is there evidence to suggest that the cross-fertilized plants have greater height?
To answer this question, consider the sign (+ or - ) of the difference in the heights of the plants in each pair by first calculating Difference = Cross – Self and then looking at the sign of the difference (+ or -).
a) If there is no difference in the cross- and self-fertilized plants in terms of height what is the probability that the cross-fertilized plant has a greater height than a self-fertilized plant grown under the same conditions, i.e. what is P(Cross – Self > 0) = P(+)? Explain your answer. (1 pts.)
b) Count how many positive differences there are and compute the probability of getting that many or more using the probability of a positive difference from part (a). (2 pts.)
c) Use the probability found in part (b) to answer the question of interest to Darwin. Explain your reasoning. (3 pts.)
3. Radioactive Gas Emissions at Prairie Island Reactor (7 pts.)
The Prairie Island nuclear power plant releases a detectable amount of radioactive gases twice a month on average.
a) Find the probability that there will be no emissions during a 3-month period. You will need to adjust so that it reflects average number of emission during a 3-month period.
(2 pts.)
b) Find the probability there will be at most four such emissions during the 3-month period. (2 pts.)
c) If during the 3-month period 12 or more emissions were detected, do you feel that there is reason to suspect the reported average figure of twice a month? Explain/justify your answer. (3 pts.)
4. Diabetes Screening Using Fasting Glucose Levels (12 pts.)
A standard test for diabetes is based on glucose levels in the blood after fasting for prescribed period. For healthy people the mean fasting glucose level is found to be 5.31 mole/liter with a standard deviation of 0.58mole/liter. For untreated diabetics the mean is 11.74, and the standard deviation is 3.50. In both groups the levels appear to be approximately Normally distributed.
To operate a simple diagnostic test based on fasting glucose levels we need to set a cutoff point, C, so that if a patient’s fasting glucose level is at least C we say they have diabetes. If it is lower, we say they do not have diabetes. Suppose we use C = 6.5.
a) What is the probability that a diabetic is correctly diagnosed as having diabetes, i.e. what is the sensitivity of the test? (2 pts.)
b) What is the probability that a nondiabetic is correctly diagnosed as not having diabetes, i.e. what is the specificity? (2 pts.)
Suppose we lower the cutoff value to C = 5.7.
c) What is the sensitivity now? (2 pts.)
d) What is the specificity now? (2 pts.)
In deciding what C to use, we have to trade off sensitivity for specificity. To do so in a reasonable way, some assessment is required of the relative “costs” of misdiagnosing a diabetic and misdiagnosing a nondiabetic. Suppose we required a 98% sensitivity.
e) What value of C gives a sensitivity of .98 or 98%? How specific is the test when C has this value? (4 pts.)