STATISTICS 2MA3

TEST #2 • 2003-03-06

Instructions

Aids permitted: any calculators, any tables and one sheet of notes (8.5" x 11", one side only).
Marks: Q1= 9, Q2 = 9, Q3 = 9, Q4 = 3, Q5 = 10.

Questions

1. (a) State Bayes’ Theorem. Give Bayes’ full name and 3 interesting facts about his life and work.
(b) Define the following terms: parameter, statistic, sampling distribution.

2. (a) Let X be the score on the upturned face after a fair 4-sided die is rolled. Find the mean and variance of X.
(b) Let Y be the total score obtained by rolling this die 9 times independently. Find the mean and variance of Y. Hence find (at least approximately) the probability that Y > 26.

3. If FEV is distributed normally with mean 4 L and standard deviation 0.5 L in adult male nonsmokers, but mean 3.5 L and standard deviation 0.6 L in adult male smokers, and a man with FEV < 3 is to be diagnosed as being a smoker, what are the sensitivity and specificity of this test? If 22% of the males in this population are smokers, what is the predictive value positive of this test?

4. The mean number of accidents at a given busy intersection has been 1.6 per week over many weeks. Last week, there were 4 accidents. Is this evidence that the mean accident rate has increased?

5. Fifteen patients were treated with a drug intended to increase heart rate The following data are the differences in heart rate before and after treatment.

4 6 5 2 5 -8 1 -2 8 0 12 13 1 0 7

(a) Is there evidence that the mean difference is not zero? Answer this question by computing a two-sided 95% confidence interval for the mean difference. State any assumptions made.
(b) Answer the same question using the number of positive differences. Why should the zero differences be omitted from this analysis?