1014SCG Statistics

PART A - Workshop Exercises
Question 1: Goodness of Fit.
The following data show the number of deaths occurring due to kicks from a horse,
per army corps per year, for 10 Prussian Army Corps over 20 years. The total number
of observations in this sample is 200.
Number of
Deaths
0
1
2
3
4
Absolute
Frequency
109
65
22
3
1
It is supposed that the distribution of deaths due to horse kicks follows a Poisson
distribution with rate 𝜆 = 0.61. If this is true the expected probabilities associated
with each category are:
Pr(Number of Deaths = 0) = 0.543
Pr(Number of Deaths = 1) = 0.331
Pr(Number of Deaths = 2) = 0.101
Pr(Number of Deaths = 3) = 0.021
Pr(Number of Deaths = 4) = 0.003.
Note: you do not need to know anything more about the Poisson distribution at this
stage, other than it would lead to the expected probabilities as stated above.
Formally test (by hand) whether the number of deaths due to horse kicks follows a
Poisson distribution with rate 𝜆 = 0.61 . Clearly state your null and alternative
hypotheses.
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Question 2: Test of Independence.
The following data were collected to test the belief that the probability of rain in
Queensland is related to the El Nino effect. The El Nino effect has been rated as
negative, zero or positive. Recordings were taken on 62 independent occasions.
Frequencies of Cross-Classified Data
Rain in Queensland El Nino Measure
negative zero positive
yes 22 6 5
no 8 8 13
(a) What is the research question in this study?
(b) State the null and alternative hypotheses you will use to test the research
question.
(c) Calculate the expected values assuming that the El Nino has no effect on
rainfall in Queensland (i.e. rainfall and El Nino are independent).
(d) Calculate the test statistic for testing the null hypothesis of independence and
determine the critical value from the chi-square table.
(e) State in clear English your conclusions from the analysis.
Question 3:
In 1982 in Western Australia, 1317 males and 854 females died of ischaemic heart
disease, 1119 males and 828 females died of cancer, 371 males and 460 females died
of cerebral vascular disease, and 346 males and 147 females died of accidents. Use
this data to test for a relationship between gender and cause of death. Clearly state
your null and alternative hypotheses.
Question 4:
Find by hand (ie using the binomial formula given in week 3/4 notes) the probability of
getting 15 ‘Heads’ when tossing a coin 20 times assuming the coin is unbiased. Then, use
both the Binomial probability generator on the web page and R (dbinom() or pbinom()) to
check your answer - see lecture notes weeks 3/4 for syntax.
Question 5:
The average rainfall in Brisbane in November is 80mm. Assuming that rainfall distribution is
normal with a standard deviation of 25mm, and that this distribution is unchanging over time,
find the probability that in November this year Brisbane will get:
(i) less than 40mm of rain;
(ii) more than 125mm of rain;
(iii) more than 75mm but less than 112mm
Use the Normal tables (ie do it by hand) and check your answers using the R function
pnorm():
Eg: > pnorm(40, mean = 80, sd = 25)
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Question 6:
The following problem is taken from Statistics Today – A Comprehensive Introduction by
Donald R. Byrkit (1987, page 262 - Benjamin/Cummings Publishing Company Inc.)
A test for right and left preference of salmon is being conducted prior to construction
of a “ladder” to aid salmon swimming upstream in getting around a dam. Temporary
artificial channels are constructed to observe whether salmon swimming upstream
prefer the left or right channel. A total of 732 salmon are observed. Of these, 348 swim
up the left channel and the remainder use the right channel. Test the hypothesis that
salmon have no left or right preference using this data.
You will need to use the normal approximation to the binomial as your sample size will be
greater than 20 – see lecture example from Week 3/4 lecture notes on the incidence of
epiphytes in eucalyptus trees.
PART B - Using R
Question 1.
Repeat questions 1, 2, and 3 in part A above using R. Syntax examples are available in the
week 2 lecture notes.
Question 2.
Redo the example from lecture notes on rainfall data using the rep() and factor() functions to
enter the variables district and season. Refer to the example given in your Week 2 lecture
notes for details.
PART C - Assessment item, 3%
Download the accompanying assessable workshop sheet in the week 4 workshop folder.
Follow the instructions on this sheet.