Sampling Distribution Theory II

Sampling Distribution of Proportion:

1. The sampling distribution of proportion is defined as:

Where x is the number of successes (values with a specified characteristic) in a sample of size n.

2. If the sampling procedure is simple random, with replacement, x is recognised as Binomial Random Variable with parameters n and π, π is the probability of success.  π can also be interpreted as the population proportion, since:

3. To determine the mean and variance of p:

Example:

A coordination team consists of seven members.  The education of each member as follows: (G = Graduate, PG = Post Graduate)

(i)                  Determine the proportion of post-graduates in the population.

(ii)                Select all possible samples of two members from the population without replacement, and compute the proportion of post-graduate members in each sample.

(iii)               Compute the mean (μ_p) and the SD (σ_p) of the sample proportion computed in (ii).

Solution:

(i) Proportion of PG in the population:

N = 7

No. of PG = 4

π = 4/7 = 0.57

(ii) No. of possible samples (without replacement) = ^NC_n = ⁷C₂ = 21 samples.

The corresponding sampling proportions are:

Sampling Distribution of Proportion

(iii) Mean ( ) and SD ( ) of sample proportion distribution:

or alternatively

The results are verified as below:

Shape of the Sampling Distribution of Proportion p:

The central limit theorem also holds for the random variable p, which states that:

(i)                  The sampling distribution of proportion p approaches a normal distribution with mean  and SD (with replacement)

(ii)                If the random sampling is without replacement and the sampling fraction , the f.p.c. must be used as below in the formula of SD:

(iii)               When n ≥ 50 and both n.π and n(1 – π) are greater than 5, the sampling distribution can be considered ‘normal’.

(iv)              When the distribution of p is normal, the following statistic will be standard normal variable:

Sampling Distribution of Difference between Two Proportions:

1. If two random samples of size n₁ and n₂ are drawn independently from two populations with proportions π₁ and π₂ the sampling distribution of (p₁ – p₂) the difference between two sample proportions, approaches normal distribution with:

as n₁ and n₂ increase.

Moreover:

will be standard normal variable.

2. For unknown π₁ and π₂, sample estimates p₁ and p₂ are used thus:

3. When the two unknown population proportions can be assumed equal, an estimated  is obtained as below:

and the estimated standard error as below:

Sampling Distribution of t:

1. If a random sample of size n is drawn from a known Normal Population with mean μ and SD σ, the sampling distribution of the sample mean  is a normal distribution with mean  and standard error , and hence z would be a standard normal variable:

2. But when the population is unknown with unknown SD σ, the value of σ is replaced the sample SD ‘S’, as given below:

Therefore, the standard error is equal to :

3. According to W.S. Gossett, the following statistics is denoted by ‘t’ instead of ‘z’, which follows another distribution known as ‘students’ t-distribution’ or simply ‘t-distribution’.
4. The sample standard deviation is given by:

In the above equation the (n – 1) is called ‘Degree of Freedom’ or simply d.f., through which we can obtain ‘t-value’ from ‘t-table’.

5. The t-distribution approaches standard normal distribution as n increases.  Typically when n > 30, the t-distribution is considered approximately standard normal.

Properties of t-distribution:

1. The t-distribution, like the standard normal, is bell shaped, unimodal and symmetrical about the mean,
2. There is a different t-distribution for every possible sample size,
3. The exact shape of t-distribution, depends on the parameter, the number of degrees of freedom, denoted by ν.
4. As the sample size increases, the shape of t-distribution becomes approximately equal to the standard normal distribution:

5. The mean and standard error of t-distribution are:

Sampling Distribution of Variances:

Population Variance:

or alternatively

Mean of sampling distribution of S² ( ):

Example:

A population consists of the following numbers: 1,3,5,7.  Find the population variance (σ²) and the mean of sampling distribution of variances ( ), if all samples are drawn with replacement of size 2 from the population.

Solution:

No. of possible samples (with replacement) = Nⁿ = 4² = 16 samples

Samples:

Means of samples:

Variances of samples:

Sampling Distribution of S²:

Pooled Estimate of Variance:

1. If random samples of size n₁ and n₂ are drawn independently from two normal populations with means μ₁ and μ₂ and variances σ₁² and σ₂², the sampling distribution of the difference between the sample means  follows a normal distribution with mean and standard error given as below:

Thus, the π will be equal to:

and it will be a standard normal variable.

2. But if σ₁² and σ₂² are unknown and equal, their estimators S₁² and S₂² are defined as:

When the σ₁² and σ₂² are replaced by the estimators S₁² and S₂² the distribution of  can be standardised provided that the samples are large (n₁ and n₂ > 30). 

3. But when samples are small, i.e., less than 30 (n₁ and n₂ ≤ 30), σ₁² and σ₂² are replaced by a single estimator known as ‘pooled variance’ denoted by S_p²:

Weighted Average of S₁² and S₂²:

Where (n₁ + n₂ – 2) is the degree of freedom.

4. With same size of samples n₁ and n₂, the estimator S_p² is the simple average of S₁² and S₂²:

5. The pooled variance S_p² assumes that the population variance is unknown and equal.  However, the same S_p² is used to replace σ₁² and σ₂² for slightly unequal population variances provided that the samples are of equal size, i.e., n₁ = n₂.
6. In both of the above situations, i.e., equal population variance and slightly unequal population variance with equal samples (i.e., n₁ = n₂), the statistic t is calculated as below:

Where S_p is pooled SD.

7. Now consider the situation where σ₁² and σ₂² are considerably different (both unknown) and it is impossible to draw samples of equal size, the statistics used in this case would be:

Where the degree of freedom ν is as follows:

Top

Home