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7.0 Sampling and Sampling Distribution 7.2 7.1 Sampling Methods Introduction to Sampling Distribution Why? Types Sampling Frame Plan 2 Institut Matematik Kejuruteraan, UniMAP BQT 173 less timeconsuming more practical WHY?? less costly less cumbersome 3 Institut Matematik Kejuruteraan, UniMAP BQT 173 Sampling Frame • listing of items that make up the population • data sources such as population lists, directories, or maps Sampling Plan • The way a sample is selected • determines the quantity of information in the sample • allow to measure the reliability or goodness of your inference 4 Institut Matematik Kejuruteraan, UniMAP BQT 173 Types of Samples Samples Nonprobability Samples Convenient Probability Samples Judgement Simple Random 5 Institut Matematik Kejuruteraan, UniMAP Systematic Stratified Cluster BQT 173 Selection of elements is left primarily to the interviewer. Easy, inexpensive, or convenient to the sample limitations- not representative of the population. Recommended for pre testing Q, generating ideas, insight @ hypotheses. Eg: a survey was conducted by one local TV stations involving a small number of housewives, white collar workers & blue collar workers. The survey attempts to elicit the respondents response towards a particular drama series aired over the channel. 6 Institut Matematik Kejuruteraan, UniMAP BQT 173 The population elements are selected based on the judgment of the researcher. From the judgment, the elements are representative of the population of interest. Eg: testing the consumers’ response towards a brand of instant coffee, Indocafe at a wholesale market. 7 Institut Matematik Kejuruteraan, UniMAP BQT 173 Definition: If a sample of n is drawn from a population of N in such a way that every possible sample of size n has the same chance of being selected, the sample obtained is called a simple random sampling. N – number of units in the population n – number of units in sample 8 Institut Matematik Kejuruteraan, UniMAP BQT 173 Do not have any bias element (every element treated equally). Target population is homogenous in nature (the units have similar characteristics) Eg: canteen operators in primary school, operators in cyber cafes, etc.. Disadvantages: Sampling frame are not updated. Sampling frame are costly to produce. Impractical for large study area. 9 Institut Matematik Kejuruteraan, UniMAP BQT 173 • Definition: a sample obtained by randomly selecting one element from the 1st k elements in the frame & every kth element there is called a 1-in-k systematic sample, with a random start. • k – interval size • k = population size sample size = N/n 10 Institut Matematik Kejuruteraan, UniMAP BQT 173 Systematic Sampling eg: Let say, there are a total of N=500 primary school canteen operators in the KlangValley in 1997 who are registered with the Ministry of Education. We required a sample of n=25 operators for a particular study. Step 1: make sure that the list is random(the name sorted alphabetically). Step 2: divide the operators into interval contain k operators. k = population size = 500/25 = 20 for every 20 operators sample size selected only one to represent that interval Step 3: 1st interval only, select r at random. Let say 7. operators with id no.7 will be 1st sample. The rest of the operators selected in remaining intervals will depend on this number. Step 4: after 7 has been selected, the remaining selection will be operators with the following id no. 11 Institut Matematik Kejuruteraan, UniMAP BQT 173 Stratified Sampling Definition: obtained by separating the population elements into non overlapping groups, called strata, & then selecting a random sample from each stratum. Large variation within the population. Eg: lecturers that can be categorized as lecturers, senior lecturers, associate prof & prof. 12 Institut Matematik Kejuruteraan, UniMAP BQT 173 Stratified Sampling Definition: obtained by separating the population elements into non overlapping groups, called strata, & then selecting a random sample from each stratum. Large variation within the population. Eg: lecturers that can be categorized as lecturers, senior lecturers, associate prof & prof. 13 Institut Matematik Kejuruteraan, UniMAP BQT 173 Cluster Sampling Definition: probability sample in which each sampling unit is a collection, @ cluster of elements. Advantages- can be applied to a large study areas - practical & economical. - cost can be reduced-interviewer only need to stay within the specific area instead travelling across of the study area. Disadvantages – higher sampling error. 14 Institut Matematik Kejuruteraan, UniMAP BQT 173 Cluster Sampling Definition: probability sample in which each sampling unit is a collection, @ cluster of elements. Advantages- can be applied to a large study areas - practical & economical. - cost can be reduced-interviewer only need to stay within the specific area instead travelling across of the study area. Disadvantages – higher sampling error. 15 Institut Matematik Kejuruteraan, UniMAP BQT 173 7.0 Sampling and Sampling Distribution 7.2 Introduction to Sampling Distribution • Sampling distribution is a probability distribution of a sample statistic based on all possible simple random sample of the same size from the same population. 7.2.1 Sampling Distribution of Mean ( ) X1 X2 X3 ... X n x n Sample mean will have a theoretical sampling distribution with mean, x E X x variance, x Var X 2 17 x N n 2 n N 1 and standard errors x N n of the sample mean X is n N 1 Institut Matematik Kejuruteraan, UniMAP BQT 173 The spread of the sampling distribution of is smaller than the spread of the corresponding population distribution. The standard deviation of the sampling distribution of decreases as the sample size increase. Consistent estimator ; the standard deviation of a sample statistics decrease as the sample size increased 18 Institut Matematik Kejuruteraan, UniMAP BQT 173 Example : The mean wage per hour for all 5000 employees who work at a large company is RM17.50 and the standard deviation is RM29.90. Let be the mean wage per hour for a random sample of certain employee selected from this company. Find the mean and standard deviation of for a sample size of 30 75 200 19 Institut Matematik Kejuruteraan, UniMAP BQT 173 Solution Population mean, x 17.50 Population standard deviation, x 29.90 (a) Mean, x x 17.50 Standard Deviation 29.90 5000 30 5000 1 30 5.4590 0.9971 5.4431 20 Institut Matematik Kejuruteraan, UniMAP BQT 173 Central Limit Theorem If we are sampling from a population that has an unknown probability distribution, the sampling distribution of the sample mean will still be approximately normal with mean and standard deviation , if the sample size is large 21 Institut Matematik Kejuruteraan, UniMAP BQT 173 Example An electronic firm has a total of 350 workers with a mean age of 37.6 years and a standard deviation of 8.3 years. If a sample of 45 workers is chosen at random from these workers, what is probability that this sample will yield an average age less than 40 years? 22 Institut Matematik Kejuruteraan, UniMAP BQT 173 Solution Population mean, x 37.6 years Population SD, x 8.3 years Population size N 350 Random sample size ,n=45 So sample mean, x 37.6 years Sample SE, x x n N n N 1 8.3 350 45 45 350 1 1.157 23 Institut Matematik Kejuruteraan, UniMAP BQT 173 Now as the sample size n 45 30 , then by Central Limit Theorem, we have N 37.6,1.157 2 X X 37.6 1.157 40 37.6 P X 40 P Z 1.157 Z P Z 2.07 0.9808 The probability that the sample will yield an average age less than 40 years is 0.9808 24 Institut Matematik Kejuruteraan, UniMAP BQT 173 7.2.2Sampling Distribution of Propotion ( ) The population and sample proportion are denoted by p and p , respectively, are calculated as, X and x p p N n where N = total number of elements in the population; X = number of elements in the population that possess a specific characteristic; n = total number of elements in the sample; and x = number of elements in the sample that possess a specific characteristic. 25 Institut Matematik Kejuruteraan, UniMAP BQT 173 Example In a recent survey of 150 household, 54 had central air conditioning. Find p and q where p is the proportion of household that have central air conditioning. Solution Since X 54 and n 150 X 54 0.36 36% n 150 n X 150 54 96 q 0.64 64% n 150 150 p 26 Institut Matematik Kejuruteraan, UniMAP BQT 173