1 SECTION 2 SIMPLE RANDOM SAMPLING 2.1 What is Simple Random Sampling? 2.1.1 Definition Simple Random Sampling --- A method of probability sampling in which a sample of n elements is randomly chosen without replacement from a population of N elements (SRSWOR vs. SRSWR) 2.1.2 One Selection Procedure for Simple Random Sampling A. Number the elements in the population (i.e., sampling frame) from 1 to N. B. Using a table of random numbers, select and record a random number between 1 and N.2 C. Select a second random number between 1 and N. If the second number is the same as the first selected number, discard it and go to the next step. If the second number is not the same as the first number, record it. D. Select a third random number between 1 and N. If this number is the same as either one of the previous numbers, discard it and go to the next step. If the number is not the same as the previous numbers, record it. E. Continue in this manner until n different numbers between 1 and N have been chosen. F. Population elements corresponding to selected numbers are an SRS sample of size n. 2.1.3 Some Statistical Notes about Simple Random Samples If we use SRS to select a sample of size n from a population of N elements: A. All possible SRS samples have the same chance of being selected.3 B. The probability that any one population element will be chosen is n/N. C. Observations taken from elements in an SRS are not statistically independent. 2.2 Estimating a Population Mean from a Simple Random Sample 2.2.1 Setting A. We have selected an SRS of size n from a population of N elements. B. We wish to use our sample to estimate the population mean per element (denoted by the symbol, Y) for some characteristic of the population. C. Examples: (1) Average annual dental care expenses for employees of a large corporation. (2) Average expenditures for prescription drugs paid by customers of a drug store chain. (3) Average height in inches of adult males students at a state university.4 2.2.2 Estimator of Y Y yy y ynynsrsniin= =+ + +==1 21 (2.1) where yi refers to the value of the i-th element selected in the sample. 2.2.3 Some Statistical Notes about ysrs A. Different SRS samples are likely to produce different values for ysrs, hence ysrs is a random variable with a sampling distribution. B. ysrs is an unbiased estimator of Y. (go to D&C; “parameter”) C. When n is large (i.e., greater than 30), the sampling distribution for ysrs closely resembles the normal distribution; this characteristic can be used when forming confidence intervals or testing hypotheses.5 2.2.4 Estimated Variance of ysrs 2srsl fv(y ) sn− =   (2.2) where f = n/N is the sampling rate and s2 is the estimated element variance for the population, calculated as sy ynn y yn ni srsiniiniin22121 121 1=−−=− −== = ( )( ) (2.3) 2.2.5 Some Statistical Notes about srsv(y ) A. The term (l-f) in formula (2.2) is called the finite population correction (fpc) which is a special adjustment to account for the fact that our sample was chosen without replacement from a finite population (i.e., an existing population of limited size). This correction factor is very nearly 1 and can be effectively ignored when the sampling rate is small (i.e., less than 0.05).6 B. Different SRS samples of the same size which are chosen from the same population are likely to produce different values for srsv(y ); hence srsv(y ) is a random variable with a sampling distribution. C. srsv(y ) is a unbiased estimator of the true variance of ysrs. D. s2 is an unbiased estimator of the population element variance. 2.2.6 Estimated Standard Error of ysrs se (ysrs) = srsv(y ) = l fn− s (2.4) where s is the square root of s2 computed by formula (2.3). 2.2.7 Confidence Interval for Y n>30 Lower Boundary: ysrs - {t}{se(ysrs)} (2.5) Upper Boundary: ysrs + {t}{se(ysrs)} (2.6)7 The value for t depends on the confidence level that we choose. For example: Confidence Level (In Percent) t 68 1.00 95 1.96 99 2.58 Interpretation: We are 95 percent sure that Y is covered by the interval whose (t = 1.96) boundaries are defined by formulas (2.5) and (2.6). 2.3 Estimating a Population Total from a Simple Random Sample 2.3.1 Setting A. We have selected an SRS of size n from a population of N elements.8 B. We wish to use our sample to estimate the population aggregate total (denoted by the symbol Yo) for some characteristic of the population. C. Examples: (1) Total combined income for all United States citizens if individual income is the characteristic of interest. (2) Total number of dental visits experienced by persons living in some small city. (3) Total dollar value of private health insurance premiums paid by workers in a large industrial plant. D. We know that YNYo=. 2.3.2 Estimator of Y nˆo osrsnisrsii 1i 1NY y Ny ynyn / N== = = = =  (2.7)9 2.3.3 Some Statistical Notes about yosrs A. Different SRS samples of the same size which are chosen from the same population are likely to produce different values for yosrs; hence yosrs is a random variable with a sampling distribution. B. yosrs is an unbiased estimator of Yo. C. The sampling distribution for yosrs is very similar to the normal distribution when n is greater than 30. 2.3.4 Estimated Variance of yosrs 2o2 2srssrsN (l f)v(y ) N v(y ) sn −= =   (2.8)10 2.3.5 Some Statistical Notes about srsv( )y A. The term (l-f) is the finite population correction (see Section 2.2.5). B. Different SRS samples of the same size which are chosen from the same population are likely to produce different values for osrsv(y ); hence osrsv(y ) is a random variable with a sampling distribution. C. osrsv(y ) is an unbiased estimator of the true variance of yosrs. 2.3.6 Estimated Standard Error of yosrs. o osrs srsl fse(y ) v(y ) Nsn−= = (2.9) where s is the square root of s2 computed by formula |(2.3).11 2.3.7 Confidence Interval for Yo ()n≥30 Lower Boundary: osrsy {t}− osrs{se y }    (2.10) Upper Boundary: osrsy {t}+ osrs{se y }    (2.11) where the value of t is determined by the confidence level (see Section