1. Sampling terminology
Sampling is an important Mathematical tool used in a wide range of fields, from piloting new drugs to testing electrical & mechanical parts to marketing.
Understanding the terminology used and the strengths & weaknesses of different sampling methods is part of S2, so some ‘English’ is required!
Eg if the Head wished to survey pupils,
A census is when all members of a population are polled.
A population is a collection of all items (of a group)
A census gives reliable information but is time consuming, expensive, and impossible if testing destroys sample (e.g. testing product lifespan).
All pupils attending the school
A sampling unit is an individual member of a population
Surveying every pupil
A sample survey is when only part of a population is polled.
The pupils
A sampling frame is a potential source of sample units.
A sample is cheaper & quicker but potentially not representative and biased.
A full list of pupils
Surveying a selection of pupils

2. WB16 A large dental practice wishes to investigate the level of satisfaction of its patients.
(a) Suggest a suitable sampling frame for the investigation.
A list of patients registered with the practice
(b) Identify the sampling units.
The patients
(c) State one advantage and one disadvantage of using a sample survey rather than a census.
Advantages: quicker, cheaper, easier
Disadvantages: potential for bias of sample

3. Read page 97 on ‘the concept of a statistic’
WB15 Explain what you understand by
(a) a population,
A collection of all items
(b) a statistic.
A random variable that is a function of the sample, that contains no unknown quantities/parameters
A researcher took a sample of 100 voters from a certain town and asked them who they would vote for in an election. The proportion who said they would vote for Dr Smith was 35%.
(c) State the population and the statistic in this case.
Population – the voters in the town
Statistic – the percentage voting for Dr Smith
(d) Explain what you understand by the sampling distribution of this statistic.
The probability distribution of those voting for Dr Smith from all possible samples of size 100

4. WB15 Explain what you understand by
(a) a population,
A collection of all items
(b) a statistic.
A random variable that is a function of the sample, that contains no unknown quantities/parameters
A researcher took a sample of 100 voters from a certain town and asked them who they would vote for in an election. The proportion who said they would vote for Dr Smith was 35%.
(c) State the population and the statistic in this case.
Population – the voters in the town
Statistic – the percentage voting for Dr Smith
(d) Explain what you understand by the sampling distribution of this statistic.
The probability distribution of those voting for Dr Smith from all possible samples of size 100

5. (a) Explain what you understand by the statistic Y.
WB17 A random sample X1, X2, ... Xn is taken from a population with unknown mean μ and unknown variance σ 2. A statistic Y is based on this sample.
(a) Explain what you understand by the statistic Y.
A random variable that is a function of the sample, that contains no unknown quantities/parameters
(b) Explain what you understand by the sampling distribution of Y.
The probability distribution of Y or the distribution of all possible values of Y
(c) State, giving a reason which of the following is not a statistic based on this sample. .
(i)
(ii)
(iii)
(ii) is not a statistic since it contains unknown parameters

6. A statistic’s sampling distribution is the set of all possible values it can take, based on the sample used, with their corresponding probabilities.
WB18 A bag contains a large number of coins.
It contains only 1p and 2p coins in the ratio 1:3.
(a) Find the mean μ and the variance σ2 of the values of this population of coins. x 1p 2p
P(X = x)
where
A random sample of size 3 is taken from the bag.
(b) List all the possible samples.
(c) Find the sampling distribution of the mean value of the samples.
Samples
Mean
Probability
Now do p102, Q7

7. Q9 A large box of coins 5p, 10p and 20p coins in the ratio 3:2:1
A random sample of 2 coins is taken from the box and their values are recorded.
List all the possible samples that can be taken.
Find the sampling distribution for the mean Y
Samples
Mean
Probability

8. Q10 A bag contains a large number of counters.
60% have a value of 6
40% have a value of 10.
A random sample of 3 counters is drawn from the bag.
Write down all the possible samples.
Find the sampling distribution for the median N
Find the sampling distribution for the mode M
Samples
Median
Mode
Probability
M & N
Probability