Sampling and Estimation 129
CHAPTER FOUR
Sampling and Estimation
Objectives
At the end of this chapter, you should be able to:
•• Explain why in many situations a sample is the only feasible way to learn something
about a population.
•• Explain the various methods of selecting a sample.
•• Distinguish between probability sampling and non-probability sampling.
•• Define and construct a sampling distribution of sample means.
•• Explain the Central Limit theorem and its importance in statistical inference.
•• Calculate confidence intervals for means and proportions.
•• Determine how large a sample should be for both means and proportions.
Fast Forward: Sampling is that part of statistical practice concerned with the selection of individual
observations intended to yield some knowledge about a population of concern, especially for the
TE X T
purposes of statistical inference.
STU D Y
Introduction
A population consists of all the items with which a particular study is concerned. A sample is a
much smaller number chosen from this population. The sample must be chosen randomly. The
data collected in the sample is used to draw inferences about the corresponding population
parameter.
The three types of distributions:
1. Population distribution
Population distribution is the distribution of the individual values of population. Its mean is
denoted by “μ”.
2. Sample distribution
It is the distribution of the individual values of a single sample. Its mean is generally written
as ‘m.’ It is extremely unlikely that it will be the same as “μ”.
3 Distribution of sample means
A sample of size n is taken from the parent population and mean of the sample is calculated.
This is repeated for a number of samples so that a population of sample means is obtained.
This population approaches a normal distribution as n increases and is the distribution of
sample means.
Definition of key terms
1. Estimate – an approximate calculation of quantity or degree or worth; an estimate of what
it would cost; a rough idea how long it would take.
, 130 q u a n t i tat i v e t e c h n i q u e s
2. Sample - a small part of something intended as representative of the whole. In statistics, a
sample is a subset of a population.
3. Probability – Probability is the likelihood or chance that something is the case or will happen.
Probability theory is used extensively in areas such as statistics, mathematics, science and
philosophy to draw conclusions about the likelihood of potential events and the underlying
mechanics of complex systems.
4. Proportion – The quotient obtained when the magnitude of a part is divided by the magnitude
of the whole; a quantity of something that is part of the whole amount or number.
5. Null hypothesis – describes some aspect of the statistical behaviour of a set of data.
This description is treated as valid unless the actual behaviour of the data contradicts this
assumption.
6. An alternative hypothesis is one that specifies that the null hypothesis is not true. The
alternative hypothesis is false when the null hypothesis is true, and true when the null
hypothesis is false. The symbol H1 is used for the alternative hypothesis.
Industry Context
With the realisation of the fact that in business time is money, dynamic technologies for
forecasting
have been a necessary toolin a wide range of managerial decisions.. In making strategic
decisions
TE X T
under uncertainty, we all make forecasts. We may not think that we are forecasting, but our
choices will be directed by our anticipation of results of our actions or inactions.
Indecision and delays are the parents of failure. For instance, budgets are intended to help
STU D Y
managers and administrators do a better job of anticipating, and hence a better job of
managing
uncertainty, by using effective forecasting and other predictive techniques.
EXAM CONTEXT
Sampling and estimation has been a popular field for examiners. The student must understand
the formulae for the previous tests to avoid confusion during an exam. Previous exam papers
where the topic has featured are:
12/02, 6/06, 12/05, 6/05, 12/04, 6/04, 6/03, 12/02, 12/00, 6/00
Fast forward: Sampling is that part of statistical practice concerned with the selection of individual
observations intended to yield knowledge about a population of concerns, especially for the
purpose of statistical inference.
, Sampling and Estimation 131
4.1 Methods of probability sampling
There is no one ‘best’ method of selecting a probability sample from a population of interest. A
method used to select a sample of invoices in a file drawer might not be the most appropriate
method to use when choosing a national sample of voters. However, all probability sampling
methods have a similar goal, namely, to allow chance to determine the items or persons to be
included in the sample.
These sampling methods include:
1. Simple random sampling
A sample is formulated in such a manner that each item or person in the population has the same
chance of being included. For instance suppose a population consists of 576 employees of Yana
Tires. A sample of 63 employees is to be selected from that population. One way is to first write all
their names, put the names in a box, mix them thoroughly then make the first selection. Repeat
this process until the 63 employees are selected.
The other method which is convenient is to use the identification number of each employee and
a table of random numbers. As the name implies, these numbers have been generated by a
TE X T
random process (in this case by a computer). Bias is therefore completely eliminated from the
selection process.
2. Systematic random sampling
STU D Y
The items or individuals of the population are arranged in some way- alphabetically, in a file
drawer by date received, or some other method. A random starting point is selected and then
every kth number of the population is selected for the sample. A systematic sample should not
be used, however, if there is a predetermined pattern to the population.
3. Stratified random sampling
A population is first divided into subgroups called strata and a sample is selected from each
stratum. After the population has been divided into strata, either a proportional or non-proportional
sample can be selected. A proportional sampling procedure requires that the number of items in
each stratum be in the same proportion as found in the population.
In a non-proportional stratified sample, the number of items studied in each stratum is
disproportionate to their number in the population. We then weight the sample results according
to the stratum’s proportion of the total population.
4. Cluster sampling
It is often employed to reduce the cost of sampling a population scattered over a large geographic
area. Suppose you want to conduct a survey to determine the views of industrialists in a state.
Selecting a random sample of industrialists in the state and personally contacting each one
would be time-consuming and very expensive. Cluster sampling would be useful by subdividing
the state into small units of regions often called primary units.
CHAPTER FOUR
Sampling and Estimation
Objectives
At the end of this chapter, you should be able to:
•• Explain why in many situations a sample is the only feasible way to learn something
about a population.
•• Explain the various methods of selecting a sample.
•• Distinguish between probability sampling and non-probability sampling.
•• Define and construct a sampling distribution of sample means.
•• Explain the Central Limit theorem and its importance in statistical inference.
•• Calculate confidence intervals for means and proportions.
•• Determine how large a sample should be for both means and proportions.
Fast Forward: Sampling is that part of statistical practice concerned with the selection of individual
observations intended to yield some knowledge about a population of concern, especially for the
TE X T
purposes of statistical inference.
STU D Y
Introduction
A population consists of all the items with which a particular study is concerned. A sample is a
much smaller number chosen from this population. The sample must be chosen randomly. The
data collected in the sample is used to draw inferences about the corresponding population
parameter.
The three types of distributions:
1. Population distribution
Population distribution is the distribution of the individual values of population. Its mean is
denoted by “μ”.
2. Sample distribution
It is the distribution of the individual values of a single sample. Its mean is generally written
as ‘m.’ It is extremely unlikely that it will be the same as “μ”.
3 Distribution of sample means
A sample of size n is taken from the parent population and mean of the sample is calculated.
This is repeated for a number of samples so that a population of sample means is obtained.
This population approaches a normal distribution as n increases and is the distribution of
sample means.
Definition of key terms
1. Estimate – an approximate calculation of quantity or degree or worth; an estimate of what
it would cost; a rough idea how long it would take.
, 130 q u a n t i tat i v e t e c h n i q u e s
2. Sample - a small part of something intended as representative of the whole. In statistics, a
sample is a subset of a population.
3. Probability – Probability is the likelihood or chance that something is the case or will happen.
Probability theory is used extensively in areas such as statistics, mathematics, science and
philosophy to draw conclusions about the likelihood of potential events and the underlying
mechanics of complex systems.
4. Proportion – The quotient obtained when the magnitude of a part is divided by the magnitude
of the whole; a quantity of something that is part of the whole amount or number.
5. Null hypothesis – describes some aspect of the statistical behaviour of a set of data.
This description is treated as valid unless the actual behaviour of the data contradicts this
assumption.
6. An alternative hypothesis is one that specifies that the null hypothesis is not true. The
alternative hypothesis is false when the null hypothesis is true, and true when the null
hypothesis is false. The symbol H1 is used for the alternative hypothesis.
Industry Context
With the realisation of the fact that in business time is money, dynamic technologies for
forecasting
have been a necessary toolin a wide range of managerial decisions.. In making strategic
decisions
TE X T
under uncertainty, we all make forecasts. We may not think that we are forecasting, but our
choices will be directed by our anticipation of results of our actions or inactions.
Indecision and delays are the parents of failure. For instance, budgets are intended to help
STU D Y
managers and administrators do a better job of anticipating, and hence a better job of
managing
uncertainty, by using effective forecasting and other predictive techniques.
EXAM CONTEXT
Sampling and estimation has been a popular field for examiners. The student must understand
the formulae for the previous tests to avoid confusion during an exam. Previous exam papers
where the topic has featured are:
12/02, 6/06, 12/05, 6/05, 12/04, 6/04, 6/03, 12/02, 12/00, 6/00
Fast forward: Sampling is that part of statistical practice concerned with the selection of individual
observations intended to yield knowledge about a population of concerns, especially for the
purpose of statistical inference.
, Sampling and Estimation 131
4.1 Methods of probability sampling
There is no one ‘best’ method of selecting a probability sample from a population of interest. A
method used to select a sample of invoices in a file drawer might not be the most appropriate
method to use when choosing a national sample of voters. However, all probability sampling
methods have a similar goal, namely, to allow chance to determine the items or persons to be
included in the sample.
These sampling methods include:
1. Simple random sampling
A sample is formulated in such a manner that each item or person in the population has the same
chance of being included. For instance suppose a population consists of 576 employees of Yana
Tires. A sample of 63 employees is to be selected from that population. One way is to first write all
their names, put the names in a box, mix them thoroughly then make the first selection. Repeat
this process until the 63 employees are selected.
The other method which is convenient is to use the identification number of each employee and
a table of random numbers. As the name implies, these numbers have been generated by a
TE X T
random process (in this case by a computer). Bias is therefore completely eliminated from the
selection process.
2. Systematic random sampling
STU D Y
The items or individuals of the population are arranged in some way- alphabetically, in a file
drawer by date received, or some other method. A random starting point is selected and then
every kth number of the population is selected for the sample. A systematic sample should not
be used, however, if there is a predetermined pattern to the population.
3. Stratified random sampling
A population is first divided into subgroups called strata and a sample is selected from each
stratum. After the population has been divided into strata, either a proportional or non-proportional
sample can be selected. A proportional sampling procedure requires that the number of items in
each stratum be in the same proportion as found in the population.
In a non-proportional stratified sample, the number of items studied in each stratum is
disproportionate to their number in the population. We then weight the sample results according
to the stratum’s proportion of the total population.
4. Cluster sampling
It is often employed to reduce the cost of sampling a population scattered over a large geographic
area. Suppose you want to conduct a survey to determine the views of industrialists in a state.
Selecting a random sample of industrialists in the state and personally contacting each one
would be time-consuming and very expensive. Cluster sampling would be useful by subdividing
the state into small units of regions often called primary units.
