&= 36471.5\\ For example, If the government wants to take a sample of 10,000 households residing in Gujarat state. We then provide the notations for two stage design. \hat{V}ar(\hat{\mu}_p) &= \dfrac{1}{n(n-1)}\sum (\bar{y}_i-\hat{\mu}_p)^2\\ A sampling at each stage, down to the penultimate stage, can be done by a variety of procedures e.g., simple random sampling, sampling with probability proportional to size (PPS), systematic sampling, and systematic sampling with PPS. \(s_i^2\) is the sample variance within the primary unit. Following this, several primary sampling units are selected at random, and from those selected primary sampling units, some smaller units, called secondary sampling units (SSU), are selected again at random. Now we have the expansion estimators from each stage. While in the multistage sampling technique, the first level is similar to that of the cluster sampling, where the clusters are formed out of the population, but further, these clusters are sub-divided into smaller targeting groups, i.e. The estimated variance of the unbiased estimator is then: \(s_u^2\) is the sample variance of \(\hat{y}_1,\ \hat{y}_2,\cdots,\ \hat{y}_{10}\). Now we have the expansion estimators from each stage. In multi-stage sampling, the fewer the stages of selection, the better and more efficient the sample would be. 9.1 - Multi-Stage Sampling: Two Stages with S.R.S at Each Stage. Find the unbiased estimator for the mean employee satisfaction score. For the example using the Restaurant Employee's Satisfaction data above, find the ratio estimator for the population mean and it estimated variance. 6.4 THE SAMPLING WORLD. Let's take a look at an example where we can compute both the estimates and their variances. We then provide the notations for two stage design. Multistage designs are used in many practical cases. See our Privacy Policy and User Agreement for details. (Chapter-2: Cluster and Multi-stage Sampling) As a result, we need to be able to figure out how this type of sampling design is implemented. The multistage sampling is a complex form of cluster sampling. Khulna University, Khulna-9208 &= 120 \cdot \dfrac{(54\times 5.50)+(48\times 6.20)+\ldots+(43\times 5.33)}{10}\\ Find the estimated variance of the unbiased estimator for the mean employee satisfaction score. \(\hat{\mu}_p=\dfrac{\sum \bar{y}_i}{n}=\dfrac{398+371.3+451.3+427.5}{4}=412.025\), \begin{align} \(s_i^2\) is the sample variance within the primary unit. \(s^2_i=\dfrac{1}{m_i-1}\sum\limits_{j=1}^{m_i}(y_{ij}-\bar{y}_i)^2\). At the second-stage, the chosen districts can be further sub-divided into the number of villages and then the sample of few villages can be taken at random. For example, in two-stage sampling: 1st stage samples n primary units; 2nd stage, for the ith primary unit, selects m i (not all) secondary units; Multistage designs are used in many practical cases. Lecturer The procedure is generalized to three or more stages and is … In two-stage sampling design, the expected values and the variances of an estimator in the sampling world are calculated taking into consideration the two stages. The estimated variance of \(\hat{\tau}\) is: \(\hat{V}ar(\hat{\tau})=N(N-n)\dfrac{s^2_u}{n}+\dfrac{N}{n}\sum\limits_{i=1}^n M_i (M_i-m_i) \dfrac{s^2_i}{m_i}\), \(s_u^2\) is the sample variance among the primary unit totals, We will discuss two possible estimators for this sampling design: unbiased estimator and ratio estimator. 9.2 - Two Stages with Primary Units Selected by Probability Proportional to Size and Secondary Units Selected with S.R.S. The third stage sample of units are chosen from the second stage sample of units, and so on. Stat-3203: Sampling Technique-II If only a sample of units is selected from each selected cluster, the method is known as two-stage sampling. 2. With multi-stage sampling, we will only select some of the units from the secondary stages. Most of the time this deals with two stages of sample with simple random sampling at each stage. If the cluster total is proportional to the cluster size, then the ratio estimate is appropriate. \hat{V}ar(\hat{\tau})&= 120 \times (120-10)\times \dfrac{3375.61}{10}+\dfrac{120}{10}\times \left(54(54-10)\dfrac{1.08^2}{10}+\ldots+43(43-9)\dfrac{1.32^2}{9}\right)\\ With multi-stage sampling, we will only select some of the units from the secondary stages. Stat 2153 Introduction to Queiueng Theory, Stat 2153 Stochastic Process and Markov chain, Stat 3203 -sampling errors and non-sampling errors, Stat 1163 -statistics in environmental science, Different kind of distance and Statistical Distance, No public clipboards found for this slide, Stat 3203 -cluster and multi-stage sampling. If n = N (all primary units are selected), it reduces to stratified random sampling. \hat{V}ar(\hat{\mu}_r)&=\dfrac{1}{M^2}\left[ \dfrac{N(N-n)}{n}\cdot \dfrac{1}{n-1}\sum\limits_{i=1}^n(\hat{y}_i-M_i\hat{r})^2+\dfrac{N}{n}\sum\limits_{i=1}^n M_i(M_i-m_i)\dfrac{s^2_i}{m_i} \right]\\ There are 36 departments in a small liberal arts college. The next thing we need is the variance. They then use simple random sampling to sample and interview about 20% of the employees in those restaurants. When does two-stage sampling reduce to cluster sampling? If we divided this by the total number of employees we would get the average score. \(s_i\) has been computed and given in the table. \hat{V}ar(\hat{\mu}_r)&=\dfrac{1}{M^2}\left[ \dfrac{N(N-n)}{n}\cdot \dfrac{1}{n-1}\sum\limits_{i=1}^n(\hat{y}_i-M_i\hat{r})^2+\dfrac{N}{n}\sum\limits_{i=1}^n M_i(M_i-m_i)\dfrac{s^2_i}{m_i} \right]\\ The cluster sampling is yet another random sampling technique wherein the population is divided into subgroups called as clusters; then few clusters are chosen randomly for the survey. &= 120 \cdot \dfrac{(54\times 5.50)+(48\times 6.20)+\ldots+(43\times 5.33)}{10}\\ Two-stage sampling includes both one-stage cluster sampling and stratified random sampling as special cases. &= 303.12\\ Initially, a population is divided into several large clusters of elements, called primary sampling units (PSU). &= 0.029\\ We use your LinkedIn profile and activity data to personalize ads and to show you more relevant ads. and secondary units selected with simple random sampling. Clipping is a handy way to collect important slides you want to go back to later. Firstly, how did you determine the number of districts to select in the first stage, inaddition for the second stage and 3rd stage, also, moreso, for the random sampling are there ratios or probabilities that are to be taken into cognizance when selecting which village, district and family to be involved. Further, the stages can be added depending on the nature of research and the size of the population under study. The next thing we need is the variance. thank you, hope to read your response. The 30x7 method is an example of what is known as a two-stage cluster sample. Multi-stage sampling (also known as multi-stage cluster sampling) is a more complex form of cluster sampling which contains two or more stages in sample selection. Multistage sampling refers to sampling plans where the sampling is carried out in stages using smaller and smaller sampling units at each stage. 1. In many surveys, the population can be divided into a set of groups of units. Since simple random sampling is used in the second stage, an unbiased estimator of the total y-value for the ith primary unit is: \(\hat{y}_i=M_i \dfrac{\sum\limits_{j=1}^{m_i}y_{ij}}{m_i}=M_i \bar{y}_i\) where \(\bar{y}_i=\dfrac{\sum\limits_{j=1}^{m_i}y_{ij}}{m_i}\). Your email address will not be published. Frequently, a hierarchy of clusters is used. compute unbiased estimator and its estimated variance for the two stage design when srs is used at each stage, compute ratio estimator and its estimated variance for the two stage design when srs is used at each stage, and.

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