Sampling & Estimation | UGC NET Commerce Preparation Course PDF Download

Sampling Distribution Overview

A sampling distribution refers to the probability distribution of a particular sample statistic (such as the mean) derived from all possible samples of a given size ‘n’ taken from a population, where the statistic is calculated for each sample.

Sampling Distribution of the Sample Mean and the Central Limit Theorem

If you collect multiple samples of size ‘n’ from a population and calculate the mean for each sample, the probability distribution of these sample means forms what is known as the ‘sampling distribution of sample means.’ The average of these sample means is represented as  , and the standard deviation of the sampling distribution of the sample means is denoted by .

Central Limit Theorem (CLT):
The central limit theorem can be simplified as follows:
When you generate a sampling distribution of sample means from a population and the sample size is sufficiently large, the distribution will tend to resemble a normal distribution.

What qualifies as a sufficiently large sample size?

• For populations that are not normally distributed, the sample size ‘n’ is generally considered sufficiently large if it is 30 or more (the "rule of 30" is a guideline but not absolute).
• If the population is normally distributed, any sample size is acceptable.

Importance of the Central Limit Theorem:
The central limit theorem indicates that for sufficiently large sample sizes, the sampling distribution of the sample means will approximate a normal distribution. This approximation improves as the sample size increases. Because of the normal distribution, the sampling distribution of the sample means can be analyzed using the standard normal variable (Z), which is essential for estimating population parameters.

Important property: Mean of the sample means  = Mean of the population (μ)

 =  σ/√n, where n is the sample size of all the samples.

So, the normal variate or the Z-score for the sampling distribution of a sample means is:

Z = () / (σ / √n)

Estimation

Estimation refers to the process of making inferences about a population based on information derived from its samples.

Types of Estimation

• Point Estimate: This involves using a statistic from a sample to estimate a population parameter. The accuracy of a point estimate depends on how well the sample represents the population. However, since sample statistics can vary across different samples, point estimates are often less reliable, which is why interval estimates are generally preferred.

• Interval Estimate: This method involves estimating a range of values (known as the confidence interval) within which a population parameter is expected to fall, along with a specified level of confidence.

The mathematics involved in interval estimate:

As discussed above, the normal variate of the sampling distribution of a sample means is:

Rearranging the equation above, you get:  

Since Z can be both positive and negative (for a random variable smaller than the mean), you have:

The equation above can be rearranged to:  

So, you can say that the population mean μ will lie between:

The formula above is used to calculate the upper and the lower limits of μ for a certain level of confidence (a certain value of Z), where the value of σ is known.
What if the value of σ is not known? In that case, you use the t-distribution.

T-distribution

Properties of T-distribution:

1. It can only be applied when the samples are drawn from a normally distributed population.

2. It is flatter than a normal distribution.
   

Degrees of freedom = Sample size - Number of unknown parameters

Here, there is only one unknown parameter: the population standard deviation. So, the degree of freedom for a t-distribution is given by ‘sample size (n) - 1’.

Standard normal variate or test statistic for t-distribution where ‘s’ is the sample standard deviation.

The formula to find the confidence interval is:

where  1-α is the confidence level associated with it.