probabilities may be obtained either from tables or from R. Incorrect numerical answers not

probabilities may be obtained either from tables or from R. Incorrect numerical answers not
supported by working will get 0. If you use R, include the R command along with the output. No
data files are required.
Total marks = 42.
1. [10 marks]
Sickle-cell disease is a hereditary medical condition affecting red blood cells. Some forms are
benign, but other forms cause health problems leading to reduced life expectancy. Sickle-cell
appears to protect against malaria, which may explain why it comes from tropical sub-Saharan
Africa where malaria is endemic. A group of African children were tested for both sickle-cell and
malaria. Out of the whole group, 23% had sickle-cell, 35% had malaria, and 6% had both.
(a) Use a contingency table to structure the above information about the incidence of sicklecell and malaria among these children. Complete all marginal totals as well as the body of
the table. [4]
(b) What proportion of children with malaria also had sickle-cell? [2]
(c) Among children who had sickle-cell, what proportion did not have malaria? [2]
(d) Among the children tested, is having malaria statistically independent of having sickle-cell?
Justify your answer. [No justification = no marks. 2]
2. [3 marks]
For a physiology class, a student is fitted with a lightweight waterproof microchip device that
measures his pulse rate at 2-hourly intervals. Consider 13 consecutive pulse-rate readings over
a 24-hour period, and let X be the number of these readings in which the pulse rate was below
90 beats per minute. Is it appropriate to treat X as an observation from a binomial distribution?
Justify your answer.
3. [13 marks]
Genetic theory predicts that, in the second generation of a cross of sweet pea plants, blossoms
will be either red or white, with each plant independently having a 70% chance of producing red
blossoms.
Out of 20 plants selected at random from the second generation of this cross, let X be the
number that have red blossoms.
(a) What is the probability distribution of X? [3]
(b) Calculate:
(i) the mean and standard deviation of X. [3]
(ii) P( X
(iii) P( 9 = X = 13) [2]
(c) Would it be unusual to find 18 red plants in the sample? Justify your answer.
[No justification = no marks. 3]
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4. [8 marks]
Arsenic occurs naturally in very low concentrations. In healthy human adults arsenic
concentration in blood is approximately Normally distributed with mean 3.72 µg/dL
(micrograms per decilitre) and standard deviation 1.18 µg/dL. For the purposes of this question,
assume that the distribution of arsenic blood concentrations is exactly as just described.
(a) What proportion of healthy adults have blood arsenic concentrations between 2 and 4
µg/dL? [3]
(b) Choosing a healthy adult at random, what is the chance that their blood arsenic
concentration exceeds 6.2 µg/dL? [2]
(c) What are the lower and upper limits of the middle 70% of blood arsenic concentrations in
healthy human adults? [3]
5. [5 marks]
A government agency is going to keep records of nitrogen levels in a lake that is mostly
surrounded by farmland. High nitrogen levels may indicate incorrect or excessive use of
fertilisers. Water samples will be taken at 5 metres depth from 15 locations on the lake and
nitrogen concentration will be measured in milligrams per litre of water. There was debate over
whether the locations to be sampled should be spread randomly over the lake’s surface, or
positioned at the mouths of water sources that feed into the lake (e.g., rivers, creeks, gullies).
(a) Briefly describe a method by which 15 locations could be selected randomly on the surface
of a lake. [Note: when referring to random selection you must describe how you would
achieve randomness.] [3]
(b) As indicated above, two sampling methods are being considered. Which of these should be
preferred if the government agency is primarily concerned with –
(i) regulating fertiliser use on farms, or [1]
(ii) environmental management, including water quality in the lake. [1]
6. [3 marks]
Calculate –
(a) the mean, and [1]
(b) the standard deviation [2]
of the random variable X, which has the following probability distribution:
X 1 2 3
P( X=x) 0.35 0.52 0.13
END OF ASSIGNMENT
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