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Mathematics & Statistics
Statistics
Examples
HT test, Q2
It is known that 20% of all farms in a particular county exceed 160
acres and that 60% of all farms in that county are owned by
persons over 50 years old. Of all farms in the county exceeding
160 acres, 55% are owned by persons over 50 years old.
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What is the probability that a randomly chosen farm in
this county both exceeds 160 acres and is owned by a
person over 50 years old?
What is the probability that a farm in this county either is
bigger than 160 acres or is owned by a person older
than 50 years old (or both)?
What is the probability that a farm in this county, owned
by a person older than 50 years old, exceeds 160
acres?
Are size of farm and age of owner in this county
independent?
HT Test, Q3
It is estimated that 45% of graduates from a particular
university graduate with a II.1 or a I (in the UK these are
called “good degrees”). Let X denote the number of
graduates with a “good degree” from a random sample
of 7 students.
 What distribution does X follow?
 Compute the probability that at most two students
graduate with a good degree.
In a particular year, 123 students graduate from the
university.
 Find the mean and standard deviation of the number of
graduates with a good degree.
 Use a normal approximation to compute the probability
that a majority of these 123 graduates obtain a good
degree.
HT Test, Q4
Betty’s is the most famous tearoom in York and they sell
tea in boxes of (nominally) 100 grams. Because of EU
legislation they take extra care and wrap, on average,
101.0 grams in each box. Assume that the weight of the
boxes is normally distributed with a standard deviation
of 0.6 grams.
 What is the probability that a box does not contain
enough tea (round to two decimals)?
 Betty’s want to reduce the probability found in (a) by
half by reducing the standard deviation. This will be
achieved by making the wrapping machine more
precise. What standard deviation should Betty’s aim for
to achieve its goal?
HT Test, Q5
A random sample of Irish workers has been obtained by
asking 20 Dublin Bus drivers outside Ringsend terminal
about their wages. We found an average gross wage of
22,350 Euros, with a standard deviation of 3,200 Euros.
In order to obtain some feeling about the accurateness
of this average, we computed a 95% confidence interval
around the mean. We do not have evidence on the
exact shape of the income distribution, but the Central
Limit Theorem tells us that the sampling distribution of
the mean is approximately normal with mean μ and
standard deviation [(3,200)2/20]1/2=715.54. Using the
formula μ±zα/2s/√n, we find a 95% confidence interval of
[22350-1.64*715.54,22350+1.64*715.54]
=[21176.51,23523.49]. So, 95% of Irish workers will
earn between 21,177 and 23,523 Euros.
Sampling distributions
Suppose that 50% of all Irish adults believe that a
major overhaul of the HSE is essential.
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What is the probability that more than 56% of a
random sample of 150 Irish adults would hold
this belief?
Estimation
A clothing store is interested in how much 3rd level
students spend on clothing during the first
month of the academic year. For a random
sample of nine students the mean expenditure
was €157.82, and the sample standard
deviation was €38.89. Assume that the
population is normal.
 Find a 95% confidence interval for the
population mean.
 What is the margin of error of this interval?
Hypothesis tests
In a random sample of 545 accountants engaged
in preparing company financial reports, 117
indicated that estimates of the cash flow were
the most difficult element of the budget to
derive.
 Test the null hypothesis that at least 25% of all
accountants find cash flow estimates the most
difficult estimates to derive at the 5%
significance level.
 What does the significance level represent?