## Statistics 2MA3 - Assignment #2

### 2002-02-16

Note: Chapt 5 Q B corrected to "10 people" on 2002-02-23, additional suggestions for Chapt 5 Q A added 2002-02-26.

## Chapter 4

A. Do the following problems from Rosner, Fundamentals of Biostatistics, 5th Edition.

• Problems 4.81 - 4.83 on p. 114
• Problems 4.87 - 4.88 on p. 115

B. The following data give the number of cases of a certain rare, non-infectous disease in a community each year for the the past ten years: 13, 8, 13, 13, 9, 10, 8, 8, 11, 5. Are these numbers consistent with the assumptions of a Poisson process, or are they over-dispersed? Assuming a Poisson process, compute the probability of (a) no cases in the next 6 weeks; (b) no cases in the next 6 weeks given that the most recent case was exactly 6 weeks ago; (c) the next case will occur in the 7th week from now.

C. In a study of herbivory, the size of each plant was measured by counting the number of leaves. There were 53 plants.

```  10  7 19  5  2  3 12  5  2  6  5  8  5  8  5  0  9  4  5  6  7 10  7  8  5
  3  4  2  3  1 10  5  3 10  4  3  3  3  2  3  2  2  4  4  2  5  4  3  5  4
  1  9  1```

How well do these data follow a Poisson distribution? Compare the sample mean and variance. Draw a histogram of the data and compare it to the graph of a Poisson distribution with the same mean. (Thanks to J. Tweedle for the data!)

## Chapter 5

A. Do the following problems from Rosner, Fundamentals of Biostatistics, 5th Edition.

• Problems 5.56 - 5.57 on pp. 150 - 151

Additional suggestions: Compare level 1, level 2, level 3 to baseline by taking differences, and by taking differences on a log scale (i.e. the log of the ratio). Which is more normal or easier to interpret? To help assess normality, plot each histogram on a probability scale and overlay a fitted normal distribution.

B. Suppose that the weight of people getting onto an elevator has mean 65 kg and standard deviation 9 kg. Each elevator trip takes 10 people. What is the probability that the total weight of people on one trip will exceed 700 kg? What is the probability that, out of five trips, at most one will exceed 700 kg?

C. Find the pdf, mean, variance and standard deviation for a uniform distribution on the interval (-0.5, 0.5). Hence find the mean, variance and approximate pdf for the sum of 12 independent observations from this distribution. Your calculator should be able to generate pseudorandom observations from a uniform distribution on the interval (0, 1). How can you use the preceeding result to generate (approximately) N(0, 1) observations on your calculator? Generate n = 5 N(0, 1 ) observations on your calculator.

D. Suppose that n observations X1, ..., Xn each have the same variance s2 but are autocorrelated in such a way that two consecutive observations have correlation r but any two non-consecutive observations are independent; that is,

Corr(Xi, Xj) = rij = r if |i - j| = 1 and rij = 0 if |i - j| > 1.

for some 0 < r <1. Apply Equation 5.13 on page 138 to show that the standard error of the sample mean is given by

sqrt{1 + 2[(n-1)/n]r} s / sqrt(n)

What happens to the standard error of the mean as r increases from 0 to 1? As n gets large? What is the importance of this result for measurements replicated under laboratory conditions? Would the model make sense for r < 0?

## Chapter 6

A. Do the following problems from Rosner, Fundamentals of Biostatistics, 5th Edition.

• Problems 6.33 - 6.34 on p. 205
• Problem 6.81 on p. 208
• Problem 6.97 - 6.98 on p. 209