7.1Sampling Methods 7.2Introduction to Sampling Distribution 7.0 Sampling and Sampling Distribution

BQT 173 Institut Matematik Kejuruteraan, UniMAP 2 SamplingWhy?FramePlanTypes

WHY?? less time- consuming less costly less cumbersome more practical BQT 173 Institut Matematik Kejuruteraan, UniMAP 3

listing of items that make up the population data sources such as population lists, directories, or maps Sampling Frame The way a sample is selected determines the quantity of information in the sample allow to measure the reliability or goodness of your inference Sampling Plan BQT 173 Institut Matematik Kejuruteraan, UniMAP 4

Types of Samples BQT 173 Institut Matematik Kejuruteraan, UniMAP 5 Samples Non- probability Samples ConvenientJudgement Probability Samples Simple Random SystematicStratifiedCluster

BQT 173 Institut Matematik Kejuruteraan, UniMAP 6 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, 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.

BQT 173 Institut Matematik Kejuruteraan, UniMAP 7 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.

BQT 173 Institut Matematik Kejuruteraan, UniMAP 8 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

BQT 173 Institut Matematik Kejuruteraan, UniMAP 9 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.

Definition: a sample obtained by randomly selecting one element from the 1 st k elements in the frame & every k th element there is called a 1-in-k systematic sample, with a random start. k – interval size k = population size sample size = N/n BQT 173 Institut Matematik Kejuruteraan, UniMAP 10

Systematic Sampling BQT 173 Institut Matematik Kejuruteraan, UniMAP 11 eg: Let say, there are a total of N=500 primary school canteen operators in the Klang Valley 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: 1 st interval only, select r at random. Let say 7. operators with id no.7 will be 1 st 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.

Stratified Sampling BQT 173 Institut Matematik Kejuruteraan, UniMAP 12 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.

Stratified Sampling BQT 173 Institut Matematik Kejuruteraan, UniMAP 13 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.

Cluster Sampling BQT 173 Institut Matematik Kejuruteraan, UniMAP 14 Definition: probability sample in which each sampling unit is a 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.

Cluster Sampling BQT 173 Institut Matematik Kejuruteraan, UniMAP 15 Definition: probability sample in which each sampling unit is a 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.

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 Introduction to Sampling Distribution 7.0 Sampling and Sampling Distribution

7.2.1Sampling Distribution of Mean ( ) BQT 173 Institut Matematik Kejuruteraan, UniMAP 17 Sample mean will have a theoretical sampling distribution with mean, variance, and standard errors of the sample mean is

BQT 173 Institut Matematik Kejuruteraan, UniMAP 18 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

Example : BQT 173 Institut Matematik Kejuruteraan, UniMAP 19 The mean wage per hour for all 5000 employees who work at a large company is RM17.50 and the standard deviation is RM 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

BQT 173 Institut Matematik Kejuruteraan, UniMAP 20 Solution Population mean, Population standard deviation, (a)Mean, Standard Deviation

Central Limit Theorem BQT 173 Institut Matematik Kejuruteraan, UniMAP 21 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

Example BQT 173 Institut Matematik Kejuruteraan, UniMAP 22 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?

Solution BQT 173 Institut Matematik Kejuruteraan, UniMAP 23 Population mean, years Population SD, years Population size Random sample size,n=45 So sample mean, years Sample SE,

BQT 173 Institut Matematik Kejuruteraan, UniMAP 24 Now as the sample size, then by Central Limit Theorem, we have The probability that the sample will yield an average age less than 40 years is

7.2.2Sampling Distribution of Propotion ( ) BQT 173 Institut Matematik Kejuruteraan, UniMAP 25 The population and sample proportion are denoted by and, respectively, are calculated as, and 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.

BQT 173 Institut Matematik Kejuruteraan, UniMAP 26 Example In a recent survey of 150 household, 54 had central air conditioning. Find and where is the proportion of household that have central air conditioning. Solution Since