Sampling and Inference. The Quality of Data and Measures
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1 Sampling and Inference The Quality of Data and Measures 1
2 Why we talk about sampling General citizen education Understand data you ll be using Understand how to draw a sample, if you need to Make statistical inferences 2
3 Cost/ benefit Why do we sample? Benefit (precision) Cost (hassle factor) N 3
4 How do we sample? Simple random sample Variant: systematic sample with a random start Stratified Cluster 4
5 Stratification Divide sample into subsamples, based on known characteristics (race, sex, religiousity, continent, department) Benefit: preserve or enhance variability 5
6 Stratification example NES Hypothetical sample N 50% N 50% White Christians 1, % % Black Christians % % White Jews % % Black Jews % % Other race/religion % % Missing 227 n.a. Total 1, % (on 1,487 valid obs.) 6
7 Cluster sampling Block HH Unit Individual 7
8 Effects of samples Obvious: influences marginals Less obvious Allows effective use of time and effort Effect on multivariate techniques Sampling of independent variable: greater precision in regression estimates Sampling on dependent variable: bias 8
9 Sampling on Independent Variable y y x x 9
10 Sampling on Dependent Variable y y x x 10
11 Sampling Consequences for Statistical Inference 11
12 Statistical Inference: Learning About the Unknown From the Known Reasoning forward: distributions of sample means, when the population mean, s.d., and n are known. Reasoning backward: learning about the population mean when only the sample, s.d., and n are known 12
13 Reasoning Forward 13
14 Exponential Distribution Example Fraction Mean = 250,000 Median=125,000 s.d. = 283,474 Min = 0 Max = 1,000, e+06 inc 14
15 Consider 10 random samples, of n = 100 apiece Sample 1 mean 253, , , ,657.3 Fraction 6 241, , , , , e+06 inc 15
16 Consider 10,000 samples of n = 100 N = 10,000 Mean = 249,993 s.d. = 28,559 Skewness = Kurtosis = 2.92 Fraction e+06 (mean) inc 16
17 Consider 1,000 samples of various sizes Fraction Fraction Fraction e+06 (mean) inc Mean =250,105 s.d.= 90,891 Skew= 0.38 Kurt= e+06 (mean) inc Mean = 250,498 s.d.= 28,297 Skew= 0.02 Kurt= e+06 (mean) inc Mean = 249,938 s.d.= 9,376 Skew= Kurt=
18 Difference of means example State 1 Mean = 250,000 Fraction e+06 inc State 2 Mean = 300,000 Fraction e+06 inc2 18
19 Take 1,000 samples of 10, of each state, and compare them First 10 samples Sample State 1 State ,410 < 365, ,571 < 243, ,574 > 438, ,374 < 557, ,934 > 189, ,400 < 284, ,115 < 210, ,885 < 333, ,678 < 314, ,725 > 152,312 19
20 1,000 samples of e+06 (mean) inc2 0 State 2 > State 1: 673 times 0 1.1e+06 (mean) inc 20
21 1,000 samples of e+06 (mean) inc e+06 (mean) inc State 2 > State 1: 909 times 21
22 1,000 samples of 1, e+06 (mean) inc e+06 (mean) inc State 2 > State 1: 1,000 times 22
23 Another way of looking at it: The distribution of Inc 2 Inc 1 n = 10 n = 100 n = 1, Fraction Fraction Fraction diff Mean = 51,845 s.d. = 124, diff Mean = 49,704 s.d. = 38, diff Mean = 49,816 s.d. = 13,932 23
24 Reasoning Backward When you know n, X, and but want to say something s, about µ 24
25 Central Limit Theorem As the sample size n increases, the distribution of the mean X of a random sample taken from practically any population approaches a normal distribution, with mean : and standard σ deviation n 25
26 Calculating Standard Errors In general: std. err. = s n 26
27 Most important standard errors Mean Proportion Diff. of 2 means Regression (slope) coeff. s n p ( 1 p) n 1 1 s p + n n 1 2 s. e. r. 1 n s x 27
28 If you know the sample mean, s.d., and n, what can you say about the population mean? In general, population mean = sample mean ± arbitrary interval standard error 28
29 If n is sufficiently large, choose the interval using the normal curve y σ 3σ 2σ σ 68% Mean 95% 99% σ 2σ 3σ 4σ 29
30 Population mean using original example (n = 10) 68% 95% 99% Sample Mean s.d. s.e. lower upper lower upper lower upper 1 311, ,392 76, , , , ,079 82, , , ,655 68, , ,767 48, ,963-20, , , , , , , , , , , , , , , ,272 50, ,971-51, , , ,256 86, , ,345 48, ,756-38, , , , , , ,817 51, ,235-57, , , ,071 62,319 64, ,435 2, ,754-59, , , ,711 40, , , , , , , , ,009 63,564 89, ,242 25, ,806-38, , , ,339 83, , ,317 55, ,908-28, ,499 30
31 Population mean using original example (n = 1000) 68% 95% 99% Sample Mean s.d. s.e. lower upper lower upper lower upper 1 238, ,492 8, , , , , , , , ,954 9, , , , , , , , ,022 8, , , , , , , , ,772 8, , , , , , , , ,343 8, , , , , , , , ,213 8, , , , , , , , ,150 9, , , , , , , , ,490 8, , , , , , , , ,994 8, , , , , , , , ,118 9, , , , , , ,262 31
32 With mean = 238,226 s.e. = 8,775, how likely is it that the true mean is 200,000 or less? Another way of asking this: y The z-ratio , , , , ,226 4σ 3σ 2σ σ Mean σ 2σ 3σ 4σ 32
33 Z (Sample mean - test value) z =, standard error in this case, (238, ,000) z = = ,775 33
34 t (when the sample is small) z (normal) distribution t-distribution z 34
35 Reading a z table 35
36 Reading a t table 36
37 Doing a t-test Q: How likely is it that the residual vote rate n 1996 was 2.5% or less?.2 Mean: s.d.: N: 1905 Fraction s. e. = s / n = = / blank96 37
38 Mean: s.d.: N: The picture s. e. = s / n = / 1905 y = t = = newz 38
39 The STATA output. ttest blank96=.025 One-sample t test Variable Obs Mean Std. Err. Std. Dev. [95% Conf. Interval] blank Degrees of freedom: 1904 Ho: mean(blank96) =.025 Ha: mean <.025 Ha: mean ~=.025 Ha: mean >.025 t = t = t = P < t = P > t = P > t =
40 Doing another t-test Q: How likely is it that the residual vote rate in 1996 equal to the rate in 1992 (I.e., blank 96 -blank 92 = 0)? Mean: s.d.: N: Fraction s. e. = s / n = = / diff
41 Mean: s.d.: N: 1448 The picture s. e. = s / n = / 1448 y = t = = newz
42 . ttest blank96=blank92 Paired t test The STATA output Variable Obs Mean Std. Err. Std. Dev. [95% Conf. Interval] blank blank diff Ho: mean(blank96 - blank92) = mean(diff) = 0 Ha: mean(diff) < 0 Ha: mean(diff) ~= 0 Ha: mean(diff) > 0 t = t = t = P < t = P > t = P > t = ttest diff9692=0 One-sample t test Variable Obs Mean Std. Err. Std. Dev. [95% Conf. Interval] diff Degrees of freedom: Ho: mean(diff9692) = 0 Ha: mean < 0 Ha: mean ~= 0 Ha: mean > 0 t = t = t = P < t = P > t = P > t =
43 Final t-test Q: Was there a relationship between residual vote and county Size in 1996? blank96 Fitted values Slope coeff: s.e.r: N: 1861 S x : s. e. = s. e. r n 1 s x = = = blank e+06 vap96_to 43
44 Calculating t t = =
45 The STATA output. reg lblank96 lvap96 Source SS df MS Number of obs = F( 1, 1859) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = lblank96 Coef. Std. Err. t P> t [95% Conf. Interval] lvap _cons
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