Lecture 5 Sampling distribution of sample mean.pptx

Objectives:
Distinguish between the distribution of population and distribution of its sample means
Explain the importance of central limit
theorem
Compute and interpret the standard error of the mean.
Sampling distribution of
sample mean:
A population is a collection or a set of measurements of interest to the researcher. For example a researcher may be interested in studying the income of households in Karachi. The measurement of interest is income of each household in Karachi and the population is a list of all households in Karachi and their incomes.
Any subset of the population is called a sample from the population. A sample of ‘n’ measurements selected from a population is said to be a random sample if every different sample of size ‘n’ from the population is equally likelyto be selected.
For the purpose of estimation of certain characteristics in the population we would like to select a random sample to be a good representative of the population.
The set of measurements in the population may be summarized by a descriptive characteristic, called a parameter. In the above example the average income of households would be the parameter.
The set of measurements in a sample may be summarized by a descriptive statistic, called a statistic . For example to estimate the average household income in Karachi, we take a random sample of the population in Karachi. The sample mean is a statistic and is an estimate of the population mean.
Because no one sample is exactly like the next , the sample mean will vary from sample to sample ,and hence is itself a random variable.
Random variables have distribution ,and since the sample mean is a random variable it must have a distribution.
If the sample mean has a normal distribution ,we can compute probabilities for specific events using the properties of the normal distribution.
Consider the population with population mean = μ
and standard deviation = σ.

Next, we take many samples of size n, calculate the mean for each one of them, and create a distribution of the sample means.

This distribution is called the Sampling Distribution of Means.

Technically, a sampling distribution of a statistic is the distribution of values of the statistic in all possible samples of the same size from the same population.
Standard error of the
mean:
The quantity σ is referred to as the standard deviation .it is a measure of spread in the population .
The quality σ/n is referred to as the standard error of the sample mean .It is a measure of spread in the distribution of mean
A very important result of statistics referring to the sampling distribution of the sample mean is the Central Limit Theorem .
Central Limit Theorem:
Consider a population with finite mean  and standard deviation . If random samples of n measurements are repeatedly drawn from the population then, when n is large, the relative frequency histogram for the sample means ( calculated from repeated samples)

1. Sampling Distribution of Sample Mean Shakir Rahman BScN , MScN, MSc Applied Psychology, PhD Nursing (Candidate) Principal & Assistant Professor Ayub International College of Nursing & AHS Peshawar Visiting Faculty Swabi College of Nursing & Health Sciences Swabi Nowshera College of Nursing & Health Sciences Nowshera

2. Objectives By the end of this session the students should be able to: Distinguish between the distribution of population and distribution of its sample means Explain the importance of central limit theorem Compute and interpret the standard error of the mean 2

3. Sampling distribution of s a mple mean A population is a collection or a set of measurements of interest to the researcher. For example a researcher may be interested in studying the income of households in Karachi. The measurement of interest is income of each household in Karachi and the population is a list of all households in Karachi and their incomes .

4. Any subset of the population is called a sample from the population. A sample of ‘n’ measurements selected from a population is said to be a random sample if every different sample of size ‘n’ from the population is equally likelyto be selected. For the purpose of estimation of certain characteristics in the population we would like to select a random sample to be a good representative of the population Sampling distribution of sample mean

5. Population and Sample Measures The set of measurements in the population may be summarized by a descriptive characteristic, called a parameter . In the above example the average income of households would be the parameter. The set of measurements in a sample may be summarized by a descriptive statistic, called a statistic . For example to estimate the average household income in Karachi, we take a random sample of the population in Karachi. The sample mean is a statistic and is an estimate of the population mean.

6. Population and Sample Measures Parameters: Mean of the Population =  Standard Deviation of the Population =  Variance of the Population =  2 Statistics (sample estimates of the parameters): Sample estimate of  = x Sample Estimate of  = s Sample Estimate of  2 = s 2

7. Normal Distributions By varying the parameters  and  , we obtain different normal distributions There are an infinite number of normal distributions

8. Sampling distribution of the mean Because no one sample is exactly like the next , the sample mean will vary from sample to sample ,and hence is itself a random variable. Random variables have distribution ,and since the sample mean is a random variable it must have a distribution. If the sample mean has a normal distribution ,we can compute probabilities for specific events using the properties of the normal distribution.

9. Population and Samples Population Sample 1 Sample 2 Sample 3 Sample 4 X 1 X 2 X 3 X 4 X 8 X 9 X10 X11 X19 X20 X21 X22 X28 X29 X30 X31

10. Sampling Distribution Consider the population with population mean = μ and standard deviation = σ. Next, we take many samples of size n, calculate the mean for each one of them, and create a distribution of the sample means. This distribution is called the Sampling Distribution of Means. Technically, a sampling distribution of a statistic is the distribution of values of the statistic in all possible samples of the same size from the same population.

11. EXAMPLE 1 The sample mean X , is to calculated from a random sample of size 2 taken from a population consisting of the five values ( $2, $3, $4, $5, $6). Find the sampling distribution x , of based on a sample of size 2. First note that the population mean,  is  = 2 + 3 + 4 + 5 + 6 = 4 5

12. EXAMPLE 1 Mean of  X values = 2.5 + 3 + 3.5 + 4 + 3.5 + 4 + 4.5+ 4.5 + 5 + 5.5 10 Possible Samples of size 2 Value of  X 2 , 3 2.5 2 , 4 3 = 4 2 , 5 3 . 5 2 , 6 4 3 , 4 3 . 5 3 , 5 4 3 , 6 4 . 5 4 , 5 4. 5 4 , 6 5 5 , 6 5. 5

13. Sampling Distribution of x  X P (  X ) 2.5 1/10 3 1/10 3.5 2/10 4 2/10 4.5 2/10 5 1/10 5.5 1/10

14. Sampling distribution of sample mean Now through the above example we have learnt that the sample mean  X has a nice mathematical property that is if you average all possible sample means which are obtained through repeating the experiment a number of times, you will obtain the population mean,  . But the variance among the sample means obtained through repeated sampling, is related to the population variance through the following formula Standard deviation of  X =  X =  /  n which will be estimated using the sample standard deviation: Standard error of  X = S /  n

15. Standard error of the mean The quantity σ is referred to as the standard deviation .it is a measure of spread in the population . The quality σ/  n is referred to as the standard error of the sample mean .It is a measure of spread in the distribution of mean A very important result of statistics referring to the sampling distribution of the sample mean is the Central Limit Theorem .

16. Central Limit Theorem Consider a population with finite mean  and standard deviation  . If random samples of n measurements are repeatedly drawn from the population then, when n is large, the relative frequency histogram for the sample means ( calculated from repeated samples) will be approximately normal (bell- shaped) with mean  and standard deviation  /  n .

17. Transforming Normal to Standard Normal Distributions Transforming Normal to Standard Normal Distributions x z   x   Note that now we use x and sigma for the p.d.f. of x.

18. The Normal Distribution Family  i z  x i   z  x    / n

19. Suppose hemoglobin level in adults is approximately normally distributed with mean 12.7 and standard deviation 2.8 – A) What proportion of adults would you expect to have HB level between 10 & 13.  z  x -   1 - 12 . 7  - 2..7 2 .8  z  x -   13 -12.7 = 0. 3 2.8 Example

20. An s w er Z -2.7 = 0.0035 Z 0.3 = 0. 1179 Area between -2.7 and 0.3 = 0.0035+0.1179= 0.1214 x 100 =12.14 % 12 .1 4 % of adults would expect to have hemoglobin level between 10 & 13.

21. Example Suppose hemoglobin level in adults is approximately normally distributed with mean 12.7 and standard deviation 2.8 – If random sample of 16 adults taken from the above population, then obtain thefollowing: The mean and standard error of sampling distribution of sample mean

22. Mean and Standard error n = 1 6 _ S.E(X) = ? _ X= ? The expected of sample mean is equal to the population mean _ S . E ( X ) =  = 12.7

23. Example Suppose hemoglobin level in a random sample of 16 adults is approximately normally distributed with mean 12.7 and standard deviation 2.8 o What proportion of adults would you expect to have HB level between 10 & 13. – We will have same result as for population or a different inference.

24. Example z  x    / n Hemoglobin level between 10 to 13 will be: Z = 10-12.7/ 0.7 = -3.85 Z -3.85= .00006 Z = 13-12.7/ 0.7 = 0.42 Z 0.42= 0.1628 Adding two probabilities 0. 00006 +0.16 28 = 0. 16286 1 6. 28 % of individuals will have Hb level between 10 & 13

25. Difference between Normal distribution Why is the normal distribution so important in the study of statistics? It’s not because things in nature are always normally distributed (although sometimes they are) It’s because of the central limit theorem—the sampling distribution of statistics (like a sample mean) often follows a normal distribution if the sample sizes are large Sampling distribution Why is sampling distribution important? If a sampling distribution has a lot of variability then if you took another sample, it’s likely you would get a very different result

26. Acknowledgements Dr Tazeen Saeed Ali RM, RM, BScN, MSc ( Epidemiology & Biostatistics), Phd (Medical Sciences), Post Doctorate (Health Policy & Planning) Associate Dean School of Nursing & Midwifery The Aga Khan University Karachi. Kiran Ramzan Ali Lalani BScN , MSc Epidemiology & Biostatistics (NICU ) Aga Khan University Hospital