7.0 Sampling and Sampling Distribution

1. 7.0 Sampling and Sampling Distribution 7.1 Sampling Methods 7.2 Introduction to Sampling Distribution

2. Institut Matematik Kejuruteraan, UniMAP

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5. Types of Samples Institut Matematik Kejuruteraan, UniMAP

6. Convenience Sampling • 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. Institut Matematik Kejuruteraan, UniMAP

7. JudgmentalSampling • 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. InstitutMatematikKejuruteraan, UniMAP

8. Simple Random Sampling (SRS) • 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 Institut Matematik Kejuruteraan, UniMAP

9. Simple Random Sampling • 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. Institut Matematik Kejuruteraan, UniMAP

10. Systematic Sampling • Definition: a sample obtained by randomly selecting one element from the 1stk elements in the frame & every kth element there is called a 1-in-k systematic sample, with a random start. • k – interval size Institut Matematik Kejuruteraan, UniMAP

11. Systematic Sampling • 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: 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. Institut Matematik Kejuruteraan, UniMAP

12. 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. Institut Matematik Kejuruteraan, UniMAP

13. Stratified Sampling Step 1: segregate population units into individual characteristics. Each unit appears in into1 group/stratum. This what meant by ‘non overlapping’. Denote the units as Nh (population stratum size). Step 2: obtain random sampling frames which are current for each stratum. Step 3: each frame select a random sample using one of method that has been discussed. Select the unit proportionately, i.e large stratum size should be represented with more units compared to small stratum size. Denote the represent unit of a particular stratum as nh (sample stratum size).

14. 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. Institut Matematik Kejuruteraan, UniMAP

15. 7.0 Sampling and Sampling Distribution

16. 7.2.1 Sampling Distribution of Mean ( ) Mean of sample mean , Mean of sampling distribution = mean of population Variance of sampling distribution Standard deviation of sampling distribution Institut Matematik Kejuruteraan, UniMAP

17. 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 n is large. Institut Matematik Kejuruteraan, UniMAP

18. Properties and shape of sampling distribution of the sample mean • n ≥ 30 , the sampling distribution of the sample mean is normally distributed • n < 30,the sampling distribution of the sample mean is normallydistributedif the sample is from the normal population and variance is known Institut Matematik Kejuruteraan, UniMAP

19. t-distributionwithn-1degreeoffreedom if the sample is from the normalpopulation but the variance is unknown The value of Z Institut Matematik Kejuruteraan, UniMAP

20. Example The amount of time required to change the oil and filter of any vehicles is normally distributed with a mean of 45 minutes and a standard deviation of 10 minutes. A random sample of 16 cars is selected. • What is the standard error of the sample mean to be? • What is the probability of the sample mean between 45 and 52 minutes? • What is the probability of the sample mean between 39 and 48 minutes? • Find the two values between the middle 95% of all sample means. Institut Matematik Kejuruteraan, UniMAP

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23. Sampling Distribution of the Sample Proportion 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 and

24. Sampling Distribution of Sample Proportion, for Infinite Population • The probability distribution of the sample proportion , is called its sampling distribution. It gives various values that can assume and their probabilities. • For the large values of n (n ≥30), the sampling distribution is very closely normally distributed. Mean of the Sample Proportion • The mean of the sample proportion, is denoted by and is equal to the population proportion, p.

25. Standard Deviation of the Sample Proportion • where p is the population proportion, and n is the sample size.

26. For a small values of n: • the population is binomial distributed, The value of Z :

27. Example 4.2 • The National Survey of Engagement shows about 87% of freshmen and seniors rate their college experience as “good” or “excellent”. Assume this result is true for the current population of freshmen and seniors. Let be the proportion of freshmen and seniors in a random sample of 900 who hold this view. Find the mean and standard deviation of .

28. Solution: • Let p the proportion of all freshmen and seniors who rate their college experience as “good” or “excellent”. Then, p = 0.87 and q = 1 – p = 1 – 0.87 = 0.13 • The mean of the sample distribution of is: The standard deviation of :

29. Sampling Distribution for the Difference between Two Means Suppose we have two populations, and which are normally distributed. and Sampling distribution for : and

30. MEAN VARIANCE

31. Therefore the distribution of can be written as:

32. Sampling Distribution for the Difference between Two Proportions Now say we have two binomial populations with proportion of successes and

33. Variance Using the Central Limit Theorem, the distribution of is

34. Example 4.4 • A certain change in a process for manufacture of component parts was considered. It was found that 75 out of 1500 items from the existing procedure were found to be defective and 80 of 2000 items from the new procedure were found to be defective. If one random sample of size 49 items were taken from the existing procedure and a random sample of 64 items were taken from the new procedure, what is the probability that • the proportion of the defective items from the new procedure exceeds the proportion of the defective items from the existing procedure? • proportions differ by at most 0.015? • the proportion of the defective items from the new procedure exceeds proportion of the defective items from the existing procedure by at least 0.02? Institut Matematik Kejuruteraan, UniMAP

35. Solution: Institut Matematik Kejuruteraan, UniMAP

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