Introduction to Sampling Distribution Lecture 25

Introduction 

to

Smpling Distribution 

Lecture 26

 Sampling Distribution

The probability distribution of the values of statistic (like mean, median, and mode) computed from all possible samples of size n selected with or without replacement is called the sampling distribution and is denoted by f(θ^) such that ∑f(θ^) =1.

Where:

 

the values of a statistic.

& 

are their respective probabilities, also known as relative frequencies.

Properties:

1. The mean of the sampling distribution of θ^ is denoted by μθ^ or E(θ^) and equal to the population mean.

 μθ^ = E(θ^) = μ

2. The standard deviation of the statistic (called standard error) is given by:

3. The shape of the sampling distribution of a statistic follows standard normal distribution.

 Sampling Distribution of Sample Mean

The probability distribution of sample means computed from all possible samples of size n selected with or without replacement is called the sampling distribution of sample mean, denoted by f(x-).

Properties

1. The mean of the sampling distribution of sample mean is given by:

2. The standard deviation of the sampling distribution of the sample mean is given by:

3. The sampling distribution of sample mean follows normal with mean 0 and standard deviation 1.

Example 7.1: A population consists of 2, 4, 6.

i. How many possible samples of size n = 2 can be selected by WR?

ii. Construct the sampling distribution of the sample mean of n = 2.

iii. Find the mean & standard deviation of the sampling distribution of sample means.

iv. Verify the following relations:

Solution: The total samples of n = 2 with replacement is given by:

Tabulation of 9 possible samples:

Possible Samples	Sample mean
2, 2	2
2, 4	3
2, 6	4
4, 2	3
4, 4	4
4, 6	5
6, 2	4
6, 4	5
6, 6	6

ii. The sampling distribution of the sample mean for n = 2.

iii. The mean & variance of the sampling distribution of the sample mean:

iv. The population mean & standard deviation:

Example 7.2: A population consists of 2, 4, 6, 8, 10

i. How many possible samples of size can be selected without replacement?

ii. Construct the sampling distribution of sample means of n = 2.

iii. Find the mean and standard deviation of the sampling distribution.

iv. Verify the relations:

Solution:

i. The total possible samples is given by:

Tabulation of possible samples and their means

ii. The sampling distribution of the sample mean:

iii. The mean and standard deviation of the sampling distribution of sample mean

iv. The population mean & standard deviation:

Example 7.3: A population consisting of 0, 3, 6, 9, 12

i. How many possible samples of size 3 can be selected without replacement?

ii. Construct the sampling distribution of the sample mean.

iii. Find the mean & standard deviation of the sampling distribution of the sample mean.

iv. Verify the following relations.

Solution:

i. The total possible samples without replacement.

Possible samples of size 3 tabulation.

ii. Sampling distribution of sample mean

iii. To find mean and standard deviation

iv. Now to compute mean & standard deviation from population.

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