We would like to describe this population. Central tendency (mean) Variability (standard deviation) ( X X ) 2 N

Transcription

1 External Validity: Assuming that there is a causal relationship in this study between the constructs of the cause and the effect, can we generalize this effect to other persons, places or times?

2 Population

3 We would like to describe this population. Central tendency (mean) Variability (standard deviation) ( X X ) 2 N

4 Population Distribution Assume a population of 10 people. How much money do they each have in their pockets? #1 = $0 #2 = $1 #3 = $2 #4 = $3 #5 = $4 #6 = $5 #7 = $6 #8 = $7 #9 = $8 #10 = $9

5 Describe the Population: central tendency measures variability o range o variance o standard deviation

6 What s the population mean? $0+$1+$2+$3+$4+$5+$6+$7+$8+$9 10 = $4.50 (Population median is also $4.50) Range = $0 to $9

7 What s the population variance? ( x x ) 2 10 (0 4.5) 2 + (1 4.5) 2 + (2 4.5) 2 + (3 4.5) 2 + (4 4.5) 2 + (5 4.5) 2 + (6 4.5) 2 + (7 4.5) 2 + (8 4.5) 2 + (9 4.5) 2 10

8 ( 4.5) 2 + ( 3.5) 2 + ( 2.5) 2 + ( 1.5) 2 + (.5) 2 + (.5) 2 + (1.5) 2 + (2.5) 2 + (3.5) 2 + (4.5) / 10 = 8.25

9 What s the population mean? = $4.5 What s the population variance? ( x x ) 2 10 Spreadsheet function: =VARP(cell:cell) = 8.25

10 What s the population standard deviation? ( x x ) = = 2.9

11 What s the population standard deviation? ( x x ) 2 10 Spreadsheet function: =STDEVP(cell:cell) = 2.9

12 We Do Not Know These Population Parameters! Solution? Estimate Them.

13 We need a sample that is large enough, representative.

14 Population Sample Distribution Assume a sample of 10 people. #1 = $0 #6 = $5 #2 = $1 #7 = $6 #3 = $2 #8 = $7 #4 = $3 #9 = $8 #5 = $4 #10 = $9

15 What s the sample mean? = $4.5 What s the sample variance? ( x x ) Spreadsheet function: =VAR(cell:cell) = 9.17

16 What s the sample standard deviation? ( x x ) Spreadsheet function: =STDEV (cell:cell) = 3.0

17 Why (n 1)? Artificially force standard deviation (and variance) to be larger. Estimating the Population standard deviation (and variance) from Sample: Unbiased estimate Conservative estimate

18 Biased vs. Unbiased Standard Deviations Sample =STDEVP Biased =STDEV Unbiased Difference ,

19 We can describe a sample. Central tendency (mean) Variability (standard deviation) ( x x ) 2 n 1 How can we use these sample statistics to estimate the population parameters?

20 The key is a characteristic of the sampling distribution:

21 Sampling Distribution An infinite number of samples of the same size.

22 With a sample size of 1, there are only 10 possible samples that can be taken, and most are not very representative.

23

24 Sampling Distribution (1) Mean = 4.5 Std. Dev. = 2.87 ± 1 Std. Dev. = 60% ± 2 Std. Dev. = 100% ± 3 Std. Dev. = 100%

25 With a sample size of 2, there are 45 possible samples that can be taken.

26

27 Sampling Distribution (2) Mean = 4.5 Std. Dev. = 1.91 ± 1 Std. Dev. = 60% ± 2 Std. Dev. = 95.5% ± 3 Std. Dev. = 100%

28 With a sample size of 3, there are 120 possible samples that can be taken.

29

30 Sampling Distribution (3) Mean = 4.5 Std. Dev. = 1.46 ± 1 Std. Dev. = 62% ± 2 Std. Dev. = 96.7% ± 3 Std. Dev. = 100%

31 With a sample size of 4, there are 210 possible samples that can be taken.

32

33 Sampling Distribution (4) Mean = 4.5 Std. Dev. = 1.17 ± 1 Std. Dev. = 65% ± 2 Std. Dev. = 96.2% ± 3 Std. Dev. = 100%

34 With a sample size of 5, there are 252 possible samples that can be taken.

35

36 Sampling Distribution (5) Mean = 4.5 Std. Dev. = 0.94 ± 1 Std. Dev. = 70% ± 2 Std. Dev. = 95.9% ± 3 Std. Dev. = 100%

37 The key is a characteristic of the sampling distribution:

38 Normal (Bell) Curve 68% 95% 99% rule

39 Sample Size Comparisons % of times the sample mean is within ±x SE of the population mean One Two Three Four Five Ave. SE NA % ± 1 SE NA % ± 2 SE NA % ± 3 SE NA

40 Sample Size Comparisons % of times the sample mean is within ±x SE of the population mean One Two Three Four Five Ave. SE NA % ± 1 SE NA % ± 2 SE NA % ± 3 SE NA

41 Sample Size Comparisons % of times the sample mean is within ±x SE of the population mean One Two Three Four Five Ave. SE NA % ± 1 SE NA % ± 2 SE NA % ± 3 SE NA

42 Sample Size Comparisons % of times the sample mean is within ±x SE of the population mean One Two Three Four Five Ave. SE NA % ± 1 SE NA % ± 2 SE NA % ± 3 SE NA

43 Sample Size Comparisons % of times the sample mean is within ±x SE of the population mean One Two Three Four Five Ave. SE NA % ± 1 SE NA % ± 2 SE NA % ± 3 SE NA

44 If you had a sampling distribution, you could predict the 68, 95 and 99% confidence intervals for the population parameter, which is why you sampled to begin with!

45 Moral: When you compute the standard deviation of a sample, which is an estimate of the population, the closer to the size of the population the sample is, the more accurate the estimate will be.

46 Standard (Sampling) Error Standard deviation of the sampling distribution Indicates precision of statistical estimate Estimated from: o Standard deviation of the sample o Sample size

47 The Result? Population estimate Variability (+/ ) Confidence (95%)

48 Terms: population (theoretical vs. accessible) sampling frame sample

49 Probability Sampling Simple Random Sampling Stratified Random Sampling Systematic Random Sampling Cluster Sampling Multi stage Sampling

50 Sample Size (Z 2 )(p)(q) / c 2 Where: Z = Z value (e.g for 95% CL) p = proportion choosing (use.5) q = 1 p c = confidence interval (e.g.,.03 = ±.03)

51 CL 95% 95% 95% 95% CI ±.02 ±.03 ±.04 ±.05 Z p q c size

52 Nonprobability Sampling Accidental, Haphazard or Convenience Sampling Purposive Sampling o Modal Sampling o Expert Sampling o Quota Sampling (proportional & nonproportional) o Heterogeneity Sampling o Snowball Sampling

53 Consider this question: What would be the best way to sample from the following populations? All Twitter Users All Tweets All Brands using Twitter