Sampling distributions chapter 6 ST 315 Nutan S. Mishra Department of Mathematics and Statistics University of South Alabama

Useful links le.htmlhttp://oak.cats.ohiou.edu/~wallacd1/ssamp le.html pling_dist/ pling_dist/

Sampling distribution In chapter 2 we defined a population parameter as a function of all the population values. Let population consists of N observations then population mean and population standard deviation are parameters For a given population, the parameters are fixed values.

Sampling distribution On the other hand if we draw a sample of size n from a population of size N, then a function of the sample values is called a statistics For example sample mean and sample standard deviation are sample statistics. Since we can draw a large number of samples from the population the value of sample statistic varies from sample to sample

Sampling distribution Since value of a sample statistic varies from sample to sample, the statistic itself is a random variable and has a probability distribution. For Example sample mean is random variable and it has a probability distribution. Example: Start with a toy example Let the population consists of 5 students who took a math quiz of 5 points. Name of the students and corresponding scores are as follows: Name of the studentABCDE Score For this population mean µ = 3.6 and standard deviation σ = 1.02

Sampling distribution Now we repeatedly draw samples of size three from the population of size 5. then the possible samples are 10 as listed below The population parameters are µ = 3.6 and s.d. σ = 1.02 Samplesample Sample values s 1A,B,C2,3,431 2A,B,D2,3,431 3A,B,E2,3, A,C,D2,4, A,C,E2,4, A,D,E2,4, B,C,D3,4, B,C,E3,4,541 9B,D,E3,4,541 10C,D,E4,4,

Sampling distribution X= score of a student in the math quiz Thus we see that the sample mean is a new random variable and has a probability distribution. Question: What is the mean of this random variable and what is its variance? xfP(x) fP( ) Population distribution Sampling distribution of sample mean

Sampling distribution Let N be the size of the population and n be the size of the sample If n/N >.05 And if n/N ≤.05

Sampling distribution of sample mean Theorem Let X be a random variable with population mean µ and population standard deviation σ. If we collect the samples of size n then the new random variable sample mean has the mean same as µ and standard deviation σ/√n We can denote them as follows:

Sampling distribution of sample mean It is easy to see that the standard deviation of sample mean decreases as the sample size increases. The mean of the sample remains unaffected with the change in sample size. Sample mean is called an estimator of the population mean. Because whenever population mean is unknown we will use sample mean in place.

Sampling distribution of sample mean P( ) From the above table when we compute the mean and variance They are

Sampling distribution of sample mean We have seen that distribution of the sample mean is derived from the distribution of x Thus distribution of x is called parent distribution. The next question is to investigate what is the relationship between the parent distribution and the sampling distribution of.

Sampling distribution of sample mean Let the distribution of x is normal with mean µ and standard deviation σ then it is equivalent to saying that Let the parent population is normal with mean µ and standard deviation σ If we draw a sample of size n from such a population then Mean of that is is equal to the mean of the population µ. Standard deviation of that is is equal to σ/√n The shape of the distribution of is normal whatever be the value of n

Sampling distribution of sample mean If X~ N(µ, σ) then ~ N ((µ, σ/√n) Where n is size of the sample drawn from the population

Central Limit Theorem For a large sample size, the sampling distribution of is approximately normal, irrespective of the shape of the population distribution. What size of the sample is considered to be large? A sample of size ≥ 30 is considered to be large.

Sampling distribution of sample mean Assume that population standard deviation σ is known If the random sample comes from a normal population, the sampling distribution of sample mean is normal regardless the size of the sample. If the shape of the parent population is not known or not normal then distribution of sample mean is approximately normal when ever n is large (≥30).(this is central limit theorem) If the shape of the parent population is not known or not normal and sample size is small then we can not say readily about the shape of sample distribution

Sampling distribution of sample mean When population standard deviation is unknown If the sample size is large the sampling distribution of sample mean is still approximately normal If the sample size is small then

About t-distribution t is a special continuous distribution Its symmetric about zero Has bell shaped curve like normal Its variance depends on the parameter is called degrees of freedom and is the only parameter of t-distribution. Variance of t approaches 1 as n  ∞ In other words t approaches Z as n  ∞ The t-values are tabulated for different values of the right tail areas and degrees of freedom

Sampling distribution of sample mean σ known n>30 Normal n<30 normal σ unknown n>30 Approx. normal n<30 t For t-distribution :assume that parent population is approximately normal