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Sampling Technique For Decision Making QQS3083

Lecture 4: Topics to Be Covered . Stratified Random SamplingIntroductionHow to draw a stratified random sampleEstimation of population mean

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Sampling Technique For Decision Making QQS3083

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    1. Sampling Technique For Decision Making QQS3083 Bidin Yatim, Associate Professor Phd Applied Statistics(Exeter, UK) MSc Industrial Mathematics (Aston, UK) BSc Mathematics & Statistics (Nottingham, UK)

    2. Lecture 4: Topics to Be Covered Stratified Random Sampling Introduction How to draw a stratified random sample Estimation of population mean & total Selecting the sample size for estimating population means and totals Allocation of the sample Estimation of population proportion Selecting the sample size and allocating the sample to estimate proportions

    3. Topics to Be Covered… Additional comments on stratified sampling Optimal rules for choosing strata Stratification after sample selection Double sampling for stratification Summary

    4. The Sampling Process

    5. Stratified random sampling Definition: A stratified random sample is one obtained by separating the population elements into non-overlapping groups, called strata, and then selecting a simple random sample from each stratum.

    6. Example Suppose an opinion poll designed to estimate the proportion of the students who favor the reintroduction of the “old bus service system” in UUM. The hostels are located on three routes. A stratified sample of students residing in hostels along these routes can be obtained by selecting a simple random sample from each route. The three routes represent three strata. If the students’ gender is believe to influence the result of the poll, the students from each route can further be divided according to gender.

    7. Why need a stratified sample? The goal of designing survey is to maximize information (or to minimize margin of error) for a fixed expenditure. Sample displaying small variability will produce small margin of error. If students from the same route of the same gender think alike on the issue, we can obtain a very accurate estimate of the interested proportion with a relatively smaller sample. Smaller sample can reduce the cost of obtaining the observations Estimates of the population parameters may be desired for subgroups (strata) of the population

    8. Example Sampling hospital patients on a certain diet to assess weight gain may be more efficient if patients are stratified by gender because men tend to weight more than women.

    9. How to select a stratified sample? CLEARLY specify the strata. Each sampling unit of the population is placed into its appropriate stratum. Select a simple random sample from each stratum using the techniques described in Lecture 3. Ensure that the selection of samples for the strata are independent by using a different sampling frame. Additional notation: L number of strata, Ni the size of stratum i, N=N1+N2+…+NL

    10. Stratified random sampling by Example Example 5.1 p120 (Scheaffer et. al. 2006). A county has 2 towns A and B and a rural area. Town A built around a factory and most households contain factory workers with school-age children. Town B an exclusive suburb of a city in the neighboring county and contains older residents with few children at home. An advert firm want to determine the average number of hours each week that households in this county watch TV. Suppose Merit of using stratified sampling: (i) Natural grouping of two towns and a rural area, according to geographical area, provide administrative convenience in selecting samples and fieldwork. (ii) Each group should have similar behavior pattern hence smaller variability and smaller error margin. (iii) Allows parameter estimates for all the groups.

    11. Estimation of a population mean and total Estimate the total of each stratum, . Population total is the sum of all strata’s total. (Equation 5.3) Estimated variance of the estimated total is given in Equation 5.4, The population mean is the population total divided by the N. Estimated variance of the estimated mean is given in Equation 5.2. and hence 95% margin of error is 2 times the estimated standard deviation of the estimator.

    12. Example 5.2 Suppose n=40 households. N1=20 from Town A, n2=8 from Town B, n3=12 from rural area. SRS selected and interview done. Results shown in Fig. 5.1. Estimate the average viewing time of all the households in the county and in Town A and B. Comment on the difference of the size of error margin in each case.

    13. Determining Sample Size for estimating population means & totals The amount of sample information depends on the sample size because the variance of the estimator decreases as n increases. If the estimator is specified to be within B units of the mean (and total) at 95% level, then the approximate sample size are given by Equation 5.6.

    14. Determining Sample Size for estimating population means & totals 2. However, the estimate of the variance of each stratum is needed before applying 5.6. Use the results of previous studies or, use the range of each stratum. By Tchebysheff’s theorem and normal distribution, the range should be 4 to 6 SD or Conduct a small-scale study of each stratum. 3. Fraction of observation allocated to each stratum ai is discuss next.

    15. Discussion of Examples 5.5 & 5.6 Ex.5.5: Estimate population mean using Eqn. 5.1., variance of each strata are obtained from previous study. Given B=2hrs, allocation fraction 1/3,1/3,1/3. Ex.5.6: Wish to stimate population total with B=400hrs. First need to determine sample size, assume each strata of equal proportion.

    16. Allocation of the sample NOTE: The best allocation provides the smallest variance at the minimum cost. After determining n, there is many ways of dividing n into each stratum size. Each may result in a difference variance. The best allocation scheme affected by: (i) Total of each stratum (ii) variability of each stratum (iii) the cost of obtaining observations in each stratum.

    17. Allocation of the sample … Large sample size for large stratum Large sample size for stratum which is less homogeneous Smaller sample for strata with high cost to keep cost of sampling at a minimum.

    18. Approx. allocation that minimize cost for a fixed value of variance of the estimator or minimize the variance at a fixed cost given by Equation 5.7, Need to approximate variance of each stratum to use Eqn 5.7. Substituting in Eqn 5.6, D is given in Eqn 5.6

    19. Example: Discuss Example 5.7: Find overall sample size and the stratum sample sizes. Bound of error given as 2hrs.

    20. Constant cost over all strata … In some cases, cost of obtaining an observation is the same for all strata. Then This method of selecting each stratum sample sizes is called Neyman allocation. Try Example 5.8 and 5.9 Two groups need to present/ discuss Example 5.10 and 5.11 next meeting …

    21. Estimation of population proportion Suppose, the advert firm in Ex. 1 would like to estimate the proportion of household that watches a particular program. Use Equation 5.13 and 5.14 for estimating population proportion and its variance:

    22. Example 5.12: The firm wants to estimate the proportion of households in the county that view Show X. Use info in Example 1 and in Table 5.12 (sample sizes 20, 8 and 12 respectively for Town A, Town B and rural area).

    23. Selecting Sample Size and Allocating the Sample to Estimate Proportion. For sample size use Equation 5.15: For allocation that minimizes cost for a it’s fixed variance or that minimizes it’s variance at fixed cost, use Equation 5.16:

    24. Example 5.13: Assume data given in Table 5.3 as the one obtained last year. The firm want to conduct a new survey, the cost for town areas is $9 per observation, and $16 each for rural household. It want to estimate the population proportion with bound of error 0.1. Find the sample size n and the strata sample sizes. Example 5.14, 5.15 and 5.16 need to be discussed during next meeting by …..

    25. Additional comments on stratified Sampling Stratified random sampling does not always produce an estimator with a smaller variance than that of SRS. Example 5.17 will illustrate this issue. To be discussed next meeting by ….

    26. Additional comments on stratified Sampling In some sample survey proble, more than one measurement is taken on each sampling unit. This situation cause complication in selecting appropriate sample size and it’s allocation. See Example 5.18

    27. An Optimal Rule for Choosing Strata Read through Section 5.9 and attemp to understand Example 5.19

    28. Stratification After Selection of the Sample Read through Section 5.10 and attempt to understand Example 5.29

    29. End of Presentation See U Later… Don’t forget to prepare for our next meeting…

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