In this article we are going to discuss about the study sampling distributions of the probability theory. The study sampling distributions of the data is the most basic concept in statistics and also this is a theoretical distribution compared to empirical distribution. In study sampling distributions of probability theory we can exercise some problems also.
The sampling distribution:
It can be defined as the distribution of the statistic for all possible samples of a given size. Each and every sampling distribution is characterized by the certain parameters. Those parameters are Mean (lambda) and variance (sigma) and this is referred as standard error. And the sampling distribution is mainly depends on the sampling size, the statics that are used in this distribution.
Sampling Distribution of Means:
Consider the population which having the mean of lambda and a standard deviation of sigma and then the sampling distribution of the mean has a mean of lambda and a standard deviation of
sigma M = sigma / sqrt(n)
Here the n is nothing but the sample size.
Standard error of the mean is nothing but the standard deviation of the sampling distribution of the mean.The mean of the sampling distribution and the mean of the population are same. So mean population has, lambda then the sampling distribution of the mean is also lambda. To refer the mean of the sampling distribution of the mean by using the symbol lambda M.
The variance of the sampling distribution of the mean is calculated as follows :
sigma^2 M=sigma^2 / N
Types of error :
There are two possibilities of error that can occur in sampling distributions. They are explained below,
1) Standard error
It can be defined as the standard deviation of the sampling distribution of statistics is termed as standard error.
2) Utility of standard error:
It will form on the basis of the testing of the hypothesis and in large sample theory it acts an important role.
Variability of a Sampling Distribution online tutoring:
It can be measured by its variance either the standard deviation of sampling distribution and it depends on the following three factors:
1) The number of observations that are present in the population.
2) The number of observations that are present in the sample.
3) The way of choosing the random sample is chosen.
If the population size is greater than the given sample size, then the sampling distribution has nearly the same sampling error, whether we sample with or without replacement.
Types of Sampling Distributions
The following are the types of the study sampling distributions of probability theory.
1) Simple Random Sampling Distribution.
2) Stratified Random Sampling Distribution.
3) Multi-Stage Sampling Distribution.
4) Cluster Sampling Distribution.
Examples of study sampling distribotions of probability:
Example 1:
Given the Sampling distribution of differences and sums S1= {5 , 10 , 12} S2 = { 4,9).
Find 1) lambda S1 2) lambda S2 3) lambda S1 + S2 .
Solution:
1) lambda U1 = (5+10+12)/3 =27/3 = 9
2) lambda U2 = (4+9)/2 =13/2 = 6.5
3) lambda U1 + U2 .
5 + 4 = 9
10+4 = 14
12 + 9=21
5+9=14
10+9=19
12+9=21
S1+S2={9,14,21,14,19,21}
lambda S1+S2= (9+14+21+14+19+21)/6=98/6=16.33
Example 2:
Find P ( >77.05) if a random sample of size 25 is drawn from an infinite population with mean lambda = 74 and sample distribution sigma= 4
Answer:
Z = (77.05-74)/4/sqrt(25)=3.8
P( > 66.75) = P (Z > 3.8)
= 0.5-0.4999
= 0.001
