1. LECTURE 3 SAMPLING THEORY EPSY 640 Texas A&M University

2. POPULATIONS finite population consists of the actual group of objects or persons, which we know is potentially countable and finite. infinite population population is a mathematical abstraction that is useful because the properties of the population are assumed or defined carefully,,

3. POPULATIONS Parameter = characteristic of the population. If a sample is drawn and the characteristic computed, it will be a statistic for the sample.

4. POPULATIONS Accessible vs. Target Populations. Target Population, the population we wish to represent. Instead, we might be able to draw from all public school grade 3 students in class during a particular week in the school year. This is our Accessible Population, the population we have access to.

6. Sampling Methods RANDOM SAMPLING –SIMPLE –STRATIFIED –MULTISTAGE –CLUSTER SYSTEMATIC SAMPLING CONVENIENCE (NONRANDOM) SAMPLING

7. RANDOM SAMPLING If every member of a population has an equal chance of being selected involves being able to define and count the population. can then use a process called randomization to select the sample

8. Table of Random Numbers Location RN 234 75 308 01 ….. 235 13 309 26 ….. 236 95 310 31 ….. 237 22 311 69 ….. 238 46 312 29 ….. 239 86 313 98 ….. 240 55 314 34 ….. 241 59 315 17 ….. In selecting a sample of 20 students from a list of of 75, a random start point was selected as shown above. The ad hoc rule was to go down the column to the bottom and up the next. Thus, children with identifiers 75 1, 13, 26, 69, 22, 46, 29, 55, 34, 59, 59, and17 have been selected within this section of the random number table. The location value allows checking and replication of a random sample selection process.

9. finite population correction fpc= 1- n/N = 1-f where n= # in sample N=# in population

10. Finding survey sample size (z  /d) 2 n = ________________________ 1 + (1/N)(z  /d) 2 z = z-score for probability for confidence interval required (usually 1.96 for.05 or 2.59 for.01)  = SD of distribution (can be 1.0 for arbitrary units) d = desired degree of error in SD units

11. Finding survey sample size- example Alpha=.05, N=1,000,000 d=.1 ,  = 1 (1.96/.1) 2 n = ________________________ 1 + (1/1000000)(1.96/.1) 2 = 19.6 2 = 384.16

12. Population SizeSample Size Required for d=.1  for  =.05 for  =.01 20 19 19 30 28 29 40 36 38 50 44 46 75 62 67 100 79 87 125 94105 150 108122 175 120138 200 132154 225142168 250151182 275 160194 300 168207 350 183229 400 196 250 500 217285 600 234315 750 254352 1000 278399 1500 306460 2000 322498 2500 333525 5000 357586 7500 365610 10000 370623 100000 383660 1000000 384663 Table 4.2: Sample sizes required for various population sizes for 95% and 99% confidence intervals

13. Mean and standard deviation for simple random sampling  (x.) =  (sample mean estimates population mean unbiasedly) V(x.) = (1/n) s 2 (1-f) (variance must be corrected) _____ s x. =  V(x.) = standard error of the mean =s m

14. -1.96s m -s m  s m 1.96s m Mean from a particular sample

15. -1.96s m -s m  s m 1.96s m Mean from a particular sample Original Data Distribution Distribution of Means

16. Confidence interval Mean  zs x. z = # SDs of normal distribution for some probability of confidence, usually.01 or.05 for real data: x.  1.96s x gives a confidence interval around the mean: –Interpretation: in 95 of 100 times we do the study, the population mean will be in the interval we construct.

17. -1.96s m -s m  s m 1.96s m Mean from a particular sample Distribution of Means Confidence interval Interpretation: in one event  is either IN or OUT of the confidence interval; for 100 intervals, it should be IN 95 times on average.

18. Stratified random sampling subpopulations, called strata. We then use simple random sampling for each stratum. We can decide to sample proportionately or disproportionately.

19. Stratified random sampling Proportionate sampling: percentage in sample is same as in population or disproportionate sampling: percentage in sample is different from that of population Example Males and Females (50% in pop.). –Proportional: 50 males, 50 females –Disproportional: 75 males, 25 females

20. Stratified random sampling Example: Ethnicity of students in District: 80% Anglo, 10% Hispanic, 5% African American, 5% Native American Proportional for 200 student sample: –160 Anglo, 20 Hispanic, 10 African-American, 10 Native American Disproportional: –50 Anglo, 50 Hispanic, 50 African-American, 50 Native American

21. Stratified random sampling Example: Ethnicity of students in District: 80% Anglo, 10% Hispanic, 5% African American, 5% Native American Proportional for 200 student sample: –160 Anglo, 20 Hispanic, 10 African-American, 10 Native American –May give poor estimates for H, AA, NA samples Disproportional: –50 Anglo, 50 Hispanic, 50 African-American, 50 Native American –Will give estimates with similar confidence intervals for all groups –may need fpc for some groups

22. Mean for stratified random sample. s x.. est =  ( N i x i. )/N i=1 Where N i = numer of cases in the population stratum i, N = total number of cases in the entire population, and s = number of strata.

23. Mean for stratified random sample- example 3 strata, N 1 =1000, N 2 =2000, N 3 =3000 X 1 = 70, X 2 = 80, X 3 = 90 s x.. est =  ( N i x i. )/N i=1 = [(1000 x 70) + (2000 x 80) + (3000 x 90) ] / 6000 = 83.33

24. SD for stratified random sample. s V(x.. est ) =  N i 2 s 2 x i. /N 2 i=1  x. =  V(x..est ), where s 2 x i. = V(x i.), the variance error of the mean using the simple random sample formula

25. SD for stratified random sample. SUBPOPULATION N I n i X.s i s m. A7750105.419 B 22950116.751 C 73850127.956 X.. est = (77 x 10 + 229 x 11 + 738 x 12)/1044 = 11.63 V(X.. est ) = (77 2 x (.419) 2 + 229 2 x (.751) 2 + 738 2 x (.956) 2 ) /1044 2 =.485 s(X.. est ) =.696 Table 4.3: Calculation of stratified sample mean and variance error of the mean

26. SD for stratified random sample. SUBPOPULATION N I n i X.s i s m. A7750105.419 B 22950116.751 C 73850127.956 s 2 m = (1/n i )s i 2 (1-f i ).419 2 = (1/50)5 2 (1-50/77).751 2 = (1/50)6 2 (1-50/229).956 2 = (1/50)7 2 (1-50/738)