Download mean weight and parasite infection (25 nest)

Dirk Metzler
SS 2014
S TATISTICS FOR EES — E XERCISE S HEET 9
1. Nestlings of the swallow Hirundo pyrrhonta are often infected by the parasite Oeciacus vicaius. For 25 nests the number of parasites per nestling and the average weight per nestling were
determined 10 days after hatching.
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mean weight and parasite infection (25 nest)
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mean weight [g]
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parasites per nestling
(a) Fit a regression line to the data by eye.
(b) Estimate the intercept and explain its biological meaning.
(c) Estimate the slope and explain its biological meaning.
(d) How precisely is the estimation of the slope? Compute the standard error.
(e) A new nest is found with 1.5 parasites per nestling. What is the prediction for the average
weight of the nestlings 10 days after hatching and how precise is this prediction?
2.
(a) Simulate normally distributed and other data sets and regard their qqnorm plots, e.g. in R
with qqnorm(rnorm(20,mean=4,sd=2)). Then look at the qqnorm plots in the file qqnorm exerc 13.pdf
and guess which of them stem from a normal distribution.
(b) The file qqnorm exerc 14.pdf contains qqnorm plots of samples from various distributions.
Sketch density plots that show how these distributions deviate from normal distributions with
the same mean and variance.
3. It may be possible that chlorinated water (as in swimming pools) damages the dental enamel.
The enamel of 100 swimmers who swam less than 6 hours per week and 100 swimmers who swam
more than 6 hours per week was surveyed.
enamel damaged
Swimming per week Yes
No
29
71
more than 6 h
less than 6 h
19
81
total
48
152
Total
100
100
200
(a) Do these observations support the hypothesis above? Perform a χ2 -test with paper, pen and
pocket calculator. Which alternative test could you perform?
(b) Estimate the proportion θ of persons with damaged enamel among the frequent-swimmers
and give a 95% confidence interval.
(c) Now assume that the same study had been performed with double sample sizes of 200 swimmers of each group. Assume further that 58 of the frequent swimmers and 38 of the others
had damaged dental enamel. Which value do you then get for the X 2 statistic and what is
your conclusion?
(d) What confidence interval for θ do you get from the values in c)?
2. A variant of Lindley’s paradox: There are two competing theories about the frequency of a
certain mutation in stone louse populations. Dr. Miller states that only 0.1% of the individuals can
have the allele. Dr. Smith claims that the relative frequency p of the allele in a population can be
any value between 0% and 100%. To decide their debate they decide to genotype 1.000 stone lice.
They find the allele only in 6 heterozygous individuals. Dr. Smith performs a statistical test with
the null hypothesis p = 0.001. Dr. Miller, who is a Bayesian, computes which theory has a higher
posterior probability given the observation. To be fair he assumes that both theories have a prior
probability of 50%. Moreover, he assumes that in Dr. Smith’s theory the prior for p is uniform
between 0% and 100%. Will the two scientists come to the same conclusion? (Note: you may have
to use R to approximate some integral by a sum.)