SAMPLING: Process of Selecting your Observations

1. SAMPLING: Process of Selecting your Observations

2. SAMPLING: Process of Selecting your Observations

3. Types of Probability Sampling: Simple (Unrestricted) Random Sampling Complex (Restricted) Probability Sampling: Some times offer more efficient alternatives to Simple Random Sampling b. Stratified Random Sampling c. Cluster Sampling a. Systematic Sampling d. Convenience Sampling e. Double Sampling

11. Sample Size Determination

12. Standard Deviation—What does it measure?

13. Income level for particular a class like this: Xs = Incomes of students in an MBA Class $6,000 $6,000 $15,000 $16,000 $39,000 $38,000 $50,000 $70,000 SX = $240,000 Average = ?x = $240,000 / 8 = $30,000

15. Life of a randomly drawn light bulb: 100 – 5 Z ? x ? 100 + 5 Z Z = 1 for 68% confidence, Z = 1.96 for 95% confidence, Z = 3for 99% confidence Formula: X = ?x + Z ?x (Where Z is an index that reflects the level of confidence/certainty with which we wish to estimate x.) SAMPLING: Process of Selecting Your Observations

16. True Population Mean = µ = Sxi / n = 45 / 10 = $4.5 Population Standard Deviation: Income of a randomly drawn person (Xi) = ?

18. Distribution of the Hypothetical Population 10 9 8 7 6 5 4 3 2 1 $0 $1 $2 $3 $4 $5 $6 $7 $8 $9

19. SAMPLING: Process of Selecting your Observations If (and only if) we know that our sample mean ( x ) comes from a normally distributed population, the same formula can be modified and applied.

20. Sampling Distribution = Frequency distribution of sample meansSampling Distribution for Samples of Size n = 2 (from our earlier population)

21. Sampling Distribution of Samples of Size n=2

22. What is SAMPLING DISTRIBUTION? Mean of a Sampling Distribution (Mean of all sample means) = µX = µX Std. Dev. of Sample Means = Standard Error = ?x

23. We will be able to say the following about the mean ( x ) of a randomly selected sample: x = mx + Z ?x Since µX = µX , substitute mx for mx : x = mx + Z ?x SAMPLING: Process of Selecting Your Observations

24. Answer: Shows the relationship between mx and x.--So, if x comes from a normal distribution, we can rewrite the formula to estimate mx based on value of x.  Question: But, is the sampling distribution (i.e., distribution of x ) always normal (so that we can use the above formula)? Let’s see it!

26. As n increases, sampling distribution (i.e., distribution of Xs) will more and more resemble a normal distribution so that for all n > 30, sampling distribution will always be normal, regardless of the distribution of the original population. 

27. True Population Mean = µ = Sxi / n = 45 / 10 = $4.5 Population Standard Deviation:

28. Distribution of the Hypothetical Population 10 9 8 7 6 5 4 3 2 1 $0 $1 $2 $3 $4 $5 $6 $7 $8 $9

29. Sampling Distribution of Samples of size n = 1 10 9 8 7 6 5 4 3 2 1 $0 $1 $2 $3 $4 $5 $6 $7 $8 $9

30. Sampling Distribution of Samples of Size n = 2

31. Sampling Distribution of Samples of Size n=2

32. Sampling Distributions of Larger Samples

33. Sampling Distributions of Samples of size 5 and 6

35. With every increase in the sample size… Distribution of sample means improves such that: ?x becomes smaller, range of x values becomes narrower and, thus, x becomes a more accurate estimate of ?. Extreme Case: What would be the largest possible sample (say, in our example)? How many such samples? What will be the sample mean? Sampling distribution (i.e., distribution of means) increasingly follows a normal distribution so thatfor n > 30, it will always be normal, regardless of the distribution shape of the original population.

36. SAMPLING: Process of Selecting Your Observations

37. SAMPLING: Process of Selecting Your Observations

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44. Assessing Resulting Accuracy/Precision of the Estimates, Given a Particular Sample Size: Suppose, we used a survey with lots of 7-point scale items, Collected data from 225 respondents, and Descriptive statistics on the data shows typical Std. Dev. on most items/variables is in the 1.3 to 1.5 range. What can we say about the precision/accuracy of our results, say, with 95% confidence/certainty? n = Z2 S2x / E2 E2 = Z2 S2 / n

45. SAMPLING: Process of Selecting your Observations Sample size determination for estimating Proportions (p): EXAMPLE: Projecting the percentage of people who would be votingfor a particular candidate in a presidential election. In such cases, dispersion is measured by = pq (instead of variance, s2) Where, p = proportion of the population that is expected to have the attribute under study, and q = (1- p), the proportion of the population that is expected NOT to have that attribute So, the sample size formula will change to: n = Z2 pq / E2 Or : NOTE: If we have no basis for judging the expected value of p, we can assume maximum variability (i.e., err on the side of overestimating the required sample size) by setting p at p=0.50 (see the example on next slid).

46. SAMPLING: Process of Selecting your Observations Sample size determination for Estimating Proportions: EXAMPLE: Suppose you are to project the percentage of potential voters who would be expected to vote for the Republican candidate in the upcoming presidential election. Suppose you have no basis for estimating/guessing what the percentage could possibly be. Also, suppose that you want to be 99% confident/certain that your margin of error would be 3% (i.e., 99% certain that your projection/estimate will be within + 3% of the actual number). What size sample will you need? n = Z2 p(1-p) / E2 Z = 3 p = 0.50 E = 0.03

47. Sample size determination for most practical situationsSource: Krejcie, R. & Morgan D. (1970). Determining Sample Size for Research Activities, Educational and Psychological Measurement, 30, 607-610.Where:  N = Population Size S = Sample Size

48. QUESTIONS OR COMMENTS ?