1. Sampling Design and Analysis MTH 494 Lecture-9
Ossam Chohan
Assistant Professor
CIIT Abbottabad

2. Review

3. Estimation of a Population Proportion
First have a look what does proportion mean?

4. Estimation of a Population Proportion
Researchers frequently interested in the portion of population possessing a specified characteristics.
E.g proportion of female voters in 2013 election.
Such situations exhibit a characteristics of the binomial experiment. ?????
Population proportion is represented by p and estimator as

5. The properties of for SRS parallel those of the sample mean y-bar if the response measurements are defined as follows: Let yi =0 if the ith element sampled does not posses the specified characteristic and yi=1 if it does.
Total number of elements in the sample is Σyi.
If we draw a SRS of size n, the sample proportion is the fraction of the elements in the sample that posses the characteristic of interest.

6. Estimators Estimator of the population proportion p:
Estimated variance of

7. Estimators
Bound on the error of estimation

8. Example
A simple random sample of n=100 college seniors was selected to estimate (i) the fraction of N=300 seniors going on to graduate school and (ii) the fraction of students that have held part-time jobs during college. Let yi and xi (i =1,2,3,…, 100) denote the responses of the ith student sampled. We will set yi =0 if the ith student does not plan to attend graduate school and yi =1 if he does. Similarly xi =0 (no part time job) and xi =1(if he has)
Estimate p1, the proportion of seniors planning to attend graduate school.
Estimate p2, the proportion of seniors who have had a part time job.

9. Data for example Student Y X 1 2 3 4 5 6 . 97 98 99 100 Summation 152 3 4 5 6 . 97 98 99
100
Summation 15 65

10. Solution

11. Solution

12. Conclusion
Thus we estimate that 0.15 or 15% of the seniors plan to attend graduate school, with a bound on error of estimation equal to (5.9%).
We estimate that 0.65 (65%) of the seniors have held a part-time job during college, with a bound on the error of estimation equal to (7.8%)

13. Sample size required to estimate p with a bound on error of estimation B
In a practical situation, we do not know p, an approximate sample size can be found by replacing p with an estimated value.
What if we put p=0.5 (if no information are provided)

14. Example
Student government leaders at a college want to conduct a survey to determine the proportion of students that favor a proposed honor code. Since interviewing N=2000 students in a reasonable length of time is almost impossible. Determine the sample size needed to estimate p with a bound on the error of estimation of magnitude B=0.05. Assume that no prior information is available to estimate p.

15. Solution

16. Practice Problem
Refer to above example, suppose that in addition to estimating the proportion of students that favor the proposed honor code, student government also want to estimate the no of students who feel the student union building adequately serves their needs. Determine the combined sample size required for a survey to estimate p1, the proportion that favor the honor code, and p2, the proportion that believes the student adequately serves its needs, with bounds on the errors of estimation of magnitude B1= 0.05 and B2=0.07. although no prior information is available to estimate p2, approximately 60% of the students believed the union adequately met their needs in a similar survey run the previous year.

17. Hint for practice problem

18. Conclusion
That is, 334 students must be interviewed to estiamte the proportion of students that favor the proposed honor code with a bound on the error of estimation of B=0.05

19. Sampling with probabilities proportional to size
So far we have discussed all cases depended on samples being a simple random sample.
In real life probabilities cannot be same for all samples.
Varying the probabilities with which different sampling units are selected is sometimes advantageous.

20. Discussion example
Suppose we wish to estimate the number of jobs openings in a city by sampling industrial firms within the city.
Many firms will be quite small while some firms will be very large.
In SRS size of firm is not taken into account.
While large firms will have more job openings as compare to small firms.
Jobs openings are highly influenced by large firms.

21. Discussion example
Therefore there should be mechanism to improve the simple random sampling by giving the large firms a greater chance to appear in the sample.
A method is called sampling with probabilities proportional to size, or pps sampling.
For a sample y1, y2, y3, … , yn, from a population of size N, Let πi= probability that yi appear in the sample
Unbiased estimators of τ and µ, along with their estimated variances and bounds on the error of estimation, are as follows:

22. Estimators
Estimator of the population Total τ
Estimation variance of

23. Bound on the error of estimation

24. Estimators
Estimator of the population mean µ:
Estimated variance of :

25. Bound on the error of estimation:

26. Important discussions
The estimators and are unbiased for any choices of πi, but it is clearly in the best interest of the experimenter to choose these πi ‘s so that the variances of the estimators are as small as possible.
Suppose for the moment that the value of yi is known for each of the N units in the population. Thus the population total τ is also known. Under these conditions we can select each unit for the sample with probability proportional to its actual measured value yi, assuming all measurements are positive. That is we can make πi= yi/τ
With π i = yi/τ for each sampled item, becomes

27. Cont….

28. Cont…..
Now to know the values y for every unit in the population before sampling is impossible. (if they were known, no sampling would be necessary). Hence the choice π i = y i /τ is not possible, but it does provide a criterion for selecting π i’s that can be used in sampling.
The best practical way to choose the π i’s is to choose them proportional to a known measurement that is highly correlated with y i.

29. Old Example with new dimension
Let us get back to the example in which population N= 4 element {1,2,3,4}
Recall that for simple random samples of size n=2, E( )= 2.5 and variance = 5/12=0.417
Suppose that we decide to sample n=2 elements with varying probabilities and choose π1= 0.1, π2= 0.1, π3=0.4, and π4= 0.4
To accomplish this sampling, we can choose a random digit from the random number table and take our first sampled element to be:

30. 1 if the random digit is 0,
2 if the random digit is 1,
3 if the random digit is 2,3,4, or 5,
4 if the random digit is 6,7,8, or 9
Note these probabilities are not exactly proportional to size but they do tend in that direction.
Listing ten possible sample

31. Sampling with varying probabilites
Sample
Probability of obtaining sample
{1,2} [2*0.1*0.1]
0.02 15
{1,3} [2*0.1*0.4]
0.08
35/4
{1,4} 10
{2,3}
55/4
{2,4}
{3,4}
0.32
{1,1}
0.01
{2,2} 20
{3,3}
0.16
15/2
{4,4}
……Where 2 is size