Sampling Distribution Theory I

Population and Sample:

  1. A ‘population’ is a well-defined group of individuals whose characteristics are to be studied.  Populations may be finite or infinite.

(a)   Finite Population: A population is said to be finite, if it consists of finite or fixed number of elements (i.e., items, objects, measurements or observations).  For example, all the university students in Pakistan, the heights of all the students enrolled in Karachi University, etc.

(b)   Infinite Population: A population is said to be infinite, if there is no limit to the number elements it can contain.  For example, the role of two dice, all the heights between 2 and 3 meters, etc.

  1. A ‘sample’ is a part of the whole selected with the object that it will represent the characteristics of the whole or population or universe.  The individuals or objects of a population or a sample may be concrete things like the motor cars produced in a company, wheat produced in a farm, or abstract things like the opinion of students about the examination system.  Thus all the students in schools, colleges and universities form population of students.  The process of selecting the sample from a population is called ‘sampling’.  A sample may be taken with replacement or without replacement:

(a)   Sampling with Replacement: If the sample is taken with replacement from a population finite or infinite, the element drawn is returned to the population before drawing the next element.

(b)   Sampling without Replacement: If the sample is taken without replacement from a finite population, the element selected is not returned to the population.

Probability Samples and Non-Probability Samples:

  1. ‘Probability samples’ are those in which every element has a known probability of being included in the sample.  Following are the probability sampling designs:

(a)   Simple Random Sampling: refers to a method of selecting a sample of a given size from a given population in such a way that all possible samples of this size which could be formed from this population have equal probabilities of selection.  It is a method in which a sample of n is selected from the population of N units such that each one of the NCn distinct samples has an equal chance of being drawn.  This method sometimes also refers to ‘lottery method’.

(b)   Stratified Random Sampling: consists of the following two steps:

(i)            The material or area to be sampled is divided into groups or classes called ‘strata’.  Items within each stratum are homogenous.

(ii)          From each stratum, a simple random sample is taken and the overall sample is obtained by combining the samples for all strata.

(c)    Systematic Sampling: is another form of sample design in which the samples are equally spaced throughout the area or population to be sampled.  For e.g., in house-to-house sampling every 10th or 20th house may be taken.  More specifically a systematic sample is obtained by taking every kth unit in the population after the units in population have been numbered or arranged in some way.

(d)   Cluster Sampling: One of the main difficulties in large scale surveys is the extensive area that may have to be covered in getting a random or stratified random sample.  It may be very expensive and lengthy task to cover the whole population in order to obtain a representative sample.  It is not possible to take a simple random or systematic sample of persons from the entire country or from within strata, since there is no such list in which all the individuals are numbered from 1 to N.  Even if such a list existed, it would be too expensive to base the enquiry on a simple random sample of persons.  Under these circumstances, it is economical to select groups called ‘clusters’ of elements from the population.  This is called ‘cluster sampling’.  The difference between a cluster and a stratum is that a stratum is expected to be homogenous and a cluster must be heterogeneous as possible.  Clusters are also known as the primary sampling units.  Cluster sampling may be consisted of:

(i)                  Single-stage Cluster Sampling,

(ii)                Sub-sampling or Two-stage Sampling, and

(iii)               Multi-stage Sampling.

  1. ‘Non-probability sampling’ designs consist of:

(a)   Judgement or Purposive Sampling: There are many situations where investigators use judgement samples to gain needed information.  For example, it may be convenient to select a random sample from a cart-load of melons.  The melons selected may be very large or very small.  The observer may use his own judgement.  This method is very useful when the sample to be drawn is small.

(b)   Quota Sampling: is widely used in opinions, market surveys, etc.  In such surveys, the interviewers are simply given quotas to be filled in from different strata, with practically no restrictions on how they are to be filled in.

Parameters and Statistic:

  1. A numerical value such as mean, median or standard deviation calculated from the population is called a ‘population parameter’ or simply a ‘parameter’.  On the other hand, a numerical value such as mean, median or SD calculated from the sample is called a ‘sample statistic’ or simply a ‘statistic’.
  2. Parameters are fixed numbers, i.e., they are constants.  Statistics very from sample to sample from the same population.
  3. In general, corresponding to each population parameter there will be a statistic to be computed from the sample.
  4. The purpose of sampling is to gather information that will be used as a basis for making generalisation about the unknown population parameters.
  5. A parameter is usually denoted by a Greek letter and a statistic is usually denoted by a Roman letter.  For e.g., the population mean is denoted by μ while the sample mean is denoted by .  Similarly, the SD of a population is denoted by σ while the sample SD is denoted by S.

Sampling and Non-Sampling Errors:

(a) Sampling Errors:

  1. The sample data deals with only a portion of the population under consideration rather than the whole population.  Because of this partial information about the population, there is always a chance of ‘errors’ or ‘discrepancies’ to exist.  This discrepancy or error is simply known as ‘sampling error’.  It is also known as ‘sampling variations’ and ‘chance variations’.
  2. Sampling error is present whenever a sample is drawn.  Mathematically, the sampling error is defined as the difference between the sample statistic and population parameter.  The conventional procedure consists of subtracting the value of parameter, θ, from that of the statistic t; that is, the sampling error, E, is:

E = t – θ

  1. The sampling errors are negative if the parameter is under estimated, and positive if it is over-estimated.
  2. The chance of sampling error can be reduced by increasing the size of the sample.

(b) Non-Sampling Errors:

  1. Such errors enter into any kind of investigation whether it is a sample or a complete census.
  2. Non-sampling errors arise from the following reasons:
  1. These errors can be controlled if the volume of data processed is small.
  2. Non-sampling errors are less significant in a sample.

Bias:

  1. It is refer to the overall or long-run tendency of the sample results to differ from the parameter in the particular way.
  2. Bias should be not be confused with sampling errors.  Mathematically, it is defined as below:

B = m – μ

Where μ is the true population value and m is the mean of the sample statistics of an infinity of samples.

  1. The bias may be positive or negative according to as m is greater or less than μ.

Precision and Accuracy:

  1. ‘Accuracy’ refers to the size of deviations from the true mean μ, whereas, the ‘precision’ refers to the size of deviation from the overall mean m obtained by repeated application of the sampling procedure.
  2. Precision is a measure of the closeness of the sample estimates to the census count taken under identical conditions and is judged in sampling theory by the variance of the estimates concerned.

Sampling Distribution:

  1. The value of a statistic varies from one sample to another even if the samples are selected from the same population.  Thus, statistic is a random variable.
  2. The distribution or probability distribution of a statistic is called a sampling distribution.  For e.g., the distribution of sample mean is a sampling distribution of mean and the distribution of the sample proportion is a sampling distribution of proportion.  The SD of the sampling distribution of a statistic is called the ‘standard error’ of the statistic.

Sampling Distribution of Mean:

From a finite population of N units with mean μ and SD σ, draw all possible random samples of size n.  Find the mean  of every sample.  Statistic is now a random variable.  Form a probability distribution of , known as ‘sampling distribution of mean’.

The sampling distribution of mean is one of the most fundamental concepts of statistical inference and it has the following properties:

  1. The mean of the sampling distribution of mean is equal to the population mean:

  1. If the sampling is done without replacement from a finite population, the standard error of mean is given by:

Where  is Finite Population Correction (f.p.c.)

 is sampling fraction

  1. When f.p.c. approaches one, the standard error of mean is simplified as:

 with replacement finite

The f.p.c. approaches one in each of the following cases:

(i)                  when the population is infinite,

(ii)                when sampling fraction  is less than 0.05, and

(iii)               when the sampling is with replacement.

Whenever, the sampling is with replacement, the population is considered infinite.  For e.g., a box contains 5 balls, when a sample is drawn with replacement, the sample size can be extended from n = 1 to n = 100 or whatever size is desired.  Hence, the population is considered to be infinite.

Mean and Standard Deviation of Sampling Distribution:

Like other distribution, the sampling distribution of  has a mean and standard deviation:

 -------------------------- Mean of sampling distribution

The standard deviation of sampling distribution of  is known as ‘standard error’ ( ).  The standard error of mean is always less than the SD of population, i.e., σ.  It depends on the size of the sample drawn.  If the sample size increases, the standard error of mean decreases and consequently the value of sample mean will be closer to the value of population mean.

 -------------------------- SD of sampling distribution

or alternatively

 ------------------------- SD of sampling distribution

No. of Possible Samples:

The number of possible samples can be calculated as below:

(i)                  When sampling is done without replacement, all possible samples = NCn

(ii)                When sampling is done with replacement, all possible samples = Nn

Example:

A population consists of following data:

1, 2, 3, 4

Suppose that a sample of size 2 is drawn ‘with replacement’.  You are required to calculate the following: 

(a)    Population mean,

(b)   Population standard deviation,

(c)    Mean of each sample,

(d)   Sampling distribution table of sample mean with replacement, and

(e)    Mean and standard deviation of sampling distribution.

Solution:

N = 4

n = 2

No. of samples (when sampling is with replacement) = Nn = 42 = 16

(a) Population Mean (μ):

(b) Population Standard Deviation (σ):

(c) Mean ( ) of Each Sample:

Samples (with replacement):

(1,1)

(2,1)

(3,1)

(4,1)

(1,2)

(2,2)

(3,2)

(4,2)

(1,3)

(2,3)

(3,3)

(4,3)

(1,4)

(2,4)

(3,4)

(4,4)

Mean ( ):

1.0

1.5

2.0

2.5

1.5

2.0

2.5

3.0

2.0

2.5

3.0

3.5

2.5

3.0

3.5

4.0

(d) Sampling Distribution:

Sampling Distribution of Sample Mean ( ) with Replacement

Frequency Distribution of

Probability Distribution of

Tally Marks

f

*  =

1.0

|

1

1.0

0.0625

1.5

||

2

1.5

0.125

2.0

|||

3

2.0

0.1875

2.5

||||

4

2.5

0.25

3.0

|||

3

3.0

0.1875

3.5

||

2

3.5

0.125

4.0

|

1

4.0

0.0625

Total

 

16

 

1

(e) Mean and standard deviation of sampling distribution:

1.0

0.0625

0.0625

–1.5

2.25

0.1406

0.0625

1.5

0.125

0.1875

–1.0

1

0.125

0.2812

2.0

0.1875

0.375

–0.5

0.25

0.0469

0.75

2.5

0.25

0.625

0

0

0

1.5625

3.0

0.1875

0.5625

0.5

0.25

0.0469

1.6875

3.5

0.125

0.4375

1.0

1

0.125

1.5312

4.0

0.0625

0.25

1.5

2.25

0.1406

1

Total

1

2.5

 

 

0.625

6.8749

Example:

Take the data of previous example and assume sampling ‘without replacement’, and compute:

(a)    Population mean,

(b)   Population standard deviation,

(c)    Mean of each sample,

(d)   Sampling distribution table of sample mean w/o replacement, and

(e)    Mean and standard deviation of sampling distribution.

Solution:

(a) and (b) Population mean and SD:

As calculated above.

(c) Mean of each sample:

No. of possible samples = NCn = 4C2 = 6 samples

Samples (without replacement):

(1,2)

(1,3)

(1,4)

(2,3)

(2,4)

(3,4)

Mean:

1.5

2

2.5

2.5

3

3.5

(d) Sampling Distribution:

Sampling Distribution of Sample Mean ( ) without replacement

f( )

1.5

1/6

0.25

–1

1

0.17

2.25

0.375

2

1/6

0.33

–0.5

0.25

0.04

4

0.666

2.5

2/6

0.84

0

0

0

6.25

2.082

3

1/6

0.5

0.5

0.25

0.04

9

1.5

3.5

1/6

0.58

1

1

0.17

12.25

2.042

Total

1

2.5

 

 

0.42

 

6.665

(e) Mean and SD of Sampling Distribution:

Sampling Distribution of the Differences of Means:

  1. Suppose we have two infinite populations I and II with means μ1 and μ2, and SD σ1 and σ2 respectively.
  2.  is the sample mean of n1 from population I and  of n2 from population II with SDs  and  respectively.
  3. From the two finite populations, we can obtain a distribution of differences of means.   is called ‘Sampling Distribution of Differences of the Means’:

Provided that  and  = 0.05

The distribution of  is normal if:

(i)                  the samples are drawn from Normal (or Symmetrical) populations, or

(ii)                n1 and n2 both are at least 30.

The distribution of ‘z’ will be standard normal:

Example:

Population I = {1, 2, 3, 4}

Population II = {3,4,5}

Samples drawn from each population with replacement:

n1 = 2

n2 = 2

Compute means of each samples, possible differences between  and , sampling distribution of , and mean and SD of sampling distribution of .

Solution:

No. of possible samples from Population I = Nn = 42 = 16 samples

Samples I:

1,1

1,2

1,3

1,4

2,1

2,2

2,3

2,4

3,1

3,2

3,3

3,4

4,1

4,2

4,3

4,4

1.0

1.5

2.0

2.5

1.5

2.0

2.5

3.0

2.0

2.5

3.0

3.5

2.5

3.0

3.5

4.0

No. of possible samples from Population II = Nn = 32 = 9 samples

Samples II:

3,3

3,4

3,5

4,3

4,4

4,5

5,3

5,4

5,5

:

3.0

3.5

4.0

3.5

4.0

4.5

4.0

4.5

5.0

Differences of Independent Sample Means

    

1

1.5

2

2.5

1.5

2

2.5

3

2

2.5

3

3.5

2.5

3

3.5

4

3

-2

-1.5

-1

-0.5

-1.5

-1

-0.5

0

-1

-0.5

0

0.5

-0.5

0

0.5

1

3.5

-2.5

-2

-1.5

-1

-2

-1.5

-1

-0.5

-1.5

-1

-0.5

0

-1

-0.5

0

0.5

4

-3

-2.5

-2

-1.5

-2.5

-2

-1.5

-1

-2

-1.5

-1

-0.5

-1.5

-1

-0.5

0

3.5

-2.5

-2

-1.5

-1

-2

-1.5

-1

-0.5

-1.5

-1

-0.5

0

-1

-0.5

0

0.5

4

-3

-2.5

-2

-1.5

-2.5

-2

-1.5

-1

-2

-1.5

-1

-0.5

-1.5

-1

-0.5

0

4.5

-3.5

-3

-2.5

-2

-3

-2.5

-2

-1.5

-2.5

-2

-1.5

-1

-2

-1.5

-1

-0.5

4

-3

-2.5

-2

-1.5

-2.5

-2

-1.5

-1

-2

-1.5

-1

-0.5

-1.5

-1

-0.5

0

4.5

-3.5

-3

-2.5

-2

-3

-2.5

-2

-1.5

-2.5

-2

-1.5

-1

-2

-1.5

-1

-0.5

5

-4

-3.5

-3

-2.5

-3.5

-3

-2.5

-2

-3

-2.5

-2

-1.5

-2.5

-2

-1.5

-1

Sampling Distribution of  with Replacement

Tally Marks

f

–4

|

1

0.00694

–0.02776

0.11104

–3.5

||||

4

0.02778

–0.09723

0.340305

–3

10

0.06945

–0.20835

0.62505

–2.5

|||

18

0.125

–0.3125

0.78125

–2

25

0.17361

–0.34722

0.69444

–1.5

|||

28

0.19444

–0.29166

0.43749

–1

25

0.17361

–0.17361

0.17361

–0.5

|||

18

0.125

–0.0625

0.03125

0

10

0.06945

0

0

0.5

||||

4

0.02778

0.01389

0.006945

1

|

1

0.00694

0.00694

0.00694

Total

 

144

1

–1.5

3.20832

Shape of the Sampling Distribution of :

The Central Limit Theorem describes the shape of the sampling distribution of mean.  The theorem states that the sampling distribution of mean is normal distribution either if the population is normal or if the sample size is more than 30.

Central limit theorem also specifies the relationship between μ and  and the relationship between σ and .

If the sampling distribution of mean is normal, we would expect 68.27%, 95.45% and 99.73% of the sample means to lie within the intervals ,  and  respectively.

Continued

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