STT 350: SURVEY SAMPLING Dr. Cuixian Chen Chapter 4: Simple Random Sampling (SRS) Elementary Survey Sampling, 7E, Scheaffer, Mendenhall, Ott and Gerow 1 Chapter 4

SRS  SRS – Every sample of size n drawn from a population of size N has the same chance of being selected.  Use table of random numbers (A.2) or computer software.  Using the table:  Assign every sampling unit a digit  Use table of random numbers to select sample

Example  In a population of N = 450, select a sample of size 10 using the table of random digits.  Assume: Table A.2 and use the second column; if we drop the last two digits of each number. (Note: we neglect the repeated numbers  Without replacement)  Starting digit value_______  Ending digit value_______  Line number started at _______  Sample digits selected for sample:

Review: Section 3.3 Summarizing Info in Populations and Samples: The Finite Population Case  If the population is infinitely large, we can assume sampling without replacement (probabilities of selecting observations are independent)  However, if population is finite, then probabilities of selecting elements will change as more elements are selected (Example: rolling a die versus selecting cards from standard 52 card deck) Chapter 3Elementary Survey Sampling, 7E, Scheaffer, Mendenhall, Ott and Gerow 4

Review: section 3.3 Sampling distribution from infinite populations  For randomly selected samples from infinite populations, mathematical properties of expected value can be used to derive the facts that:  It can also be shown that the variance of the sample mean can be estimated unbiasedly by: Chapter 3Elementary Survey Sampling, 7E, Scheaffer, Mendenhall, Ott and Gerow 5

Estimating population average with SRS chosen w/o replacement in Finite Population (N)  We use (  y i /n) to estimate  ( is an unbiased estimator of  )  We use s 2 to estimate  2 (unbiased estimator)  From previous, we know that V( ) =  2 /n (infinite population….or extremely large)  If finite population, then V( ) = ( (N-n)/(N-1)) (  2 /n)  When we replace  2 by s 2, this becomes estimated variance of y-bar = (1-(n/N))(s 2 /n)

Bound on the error of estimation for with SRS chosen w/o replacement in Finite Population (N)  For C=95%, use 2 standard errors as our bound (think of MOE), we have 2sqrt( (1-(n/N))(s 2 /n)).  When can the finite population correction (fpc) be dropped? A good rule of thumb is when (1-n/N) > 0.95  Want data to be approximately normal (sometimes transformations can be used…..the log transformation is one of the most popular transformations)

Estimating population average with SRS chosen w/o replacement in Finite Population (N) Chapter 4Elementary Survey Sampling, 7E, Scheaffer, Mendenhall, Ott and Gerow 8

Eg4.2, page 85  a) Estimate mu;  b) For C=95%, find the bound on the error of estimation for mu;  c) Find 95% CI for mu. Chapter 4Elementary Survey Sampling, 7E, Scheaffer, Mendenhall, Ott and Gerow 9

EX4.16, page 104 Chapter 4Elementary Survey Sampling, 7E, Scheaffer, Mendenhall, Ott and Gerow 10  a) Estimate mu;  b) For C=95%, find the bound on the error of estimation for mu;  c) Find 95% CI for mu.

Estimating population total using SRS chosen w/o replacement in Finite Population (N)  Since a SRS assumes all observations have an equally likely chance to be selected, we set  i to be  i = n/N  We use  -hat to estimate  ( =  y i /  i =N*y-bar is an unbiased estimator of  )  Therefore, for finite population, V(  ) = N 2 ( (N-n)/(N-1)) (  2 /n)  When we replace  2 by s 2, this becomes  estimated variance of  = N 2 (1-(n/N))(s 2 /n)

Estimating population total using SRS chosen w/o replacement in Finite Population (N)

Bound on the error of estimation  For C=95%, use 2 standard errors as our bound (think of MOE), we have 2sqrt( N 2 (1-(n/N))(s 2 /n))  Normality is still important here!! (transform if necessary….i.e. small sample size and skewed data)

Eg4.4, page 87 Chapter 4Elementary Survey Sampling, 7E, Scheaffer, Mendenhall, Ott and Gerow 14  a) Estimate the Total  ;  b) For C=95%, find the bound on the error of estimation for  ;  c) Find 95% CI for the Total .

EX4.17, page 104 Chapter 4Elementary Survey Sampling, 7E, Scheaffer, Mendenhall, Ott and Gerow 15  a) Estimate the Total  ;  b) For C=95%, find the bound on the error of estimation for  ;  c) Find 95% CI for the Total . From above

Selecting Sample Size for  using SRS chosen w/o replacement in Finite Population (N)  Use variance of y-bar: V(y-bar) = ( (N-n)/(N-1)) (  2 /n).  Set B = 2sqrt(V(y-bar)), which is B = 2sqrt(( (N-n)/(N-1)) (  2 /n) ) and solve for n. ….which yields n = (N  2 )/((N-1)D+  2 ) where D=B 2 /4  Since  2 is usually not known, estimate it with s 2 (or s is approximately range/4)

Selecting Sample Size for  using SRS chosen w/o replacement in Finite Population (N) Chapter 4Elementary Survey Sampling, 7E, Scheaffer, Mendenhall, Ott and Gerow 17

Eg4.5, page 89 Chapter 4Elementary Survey Sampling, 7E, Scheaffer, Mendenhall, Ott and Gerow 18  a) Find sample size n.

EX , page 106 Chapter 4Elementary Survey Sampling, 7E, Scheaffer, Mendenhall, Ott and Gerow 19  a) Based on the information from EX4.23, find sample size n for EX4.24.

Selecting Sample Size for  using SRS chosen w/o replacement in Finite Population (N)  Set B = 2sqrt(N 2 V(y-bar)), which is B = 2sqrt(N 2 ( (N-n)/(N-1)) (  2 /n) ) and solve for n ….which yields n = (N  2 )/((N-1)D+  2 ) where D=B 2 /(4N 2 )  Since  2 is usually not known, estimate it with s 2 (or s is approximately range/4)

Selecting Sample Size for  using SRS chosen w/o replacement in Finite Population (N) Chapter 4Elementary Survey Sampling, 7E, Scheaffer, Mendenhall, Ott and Gerow 21

Eg4.6, page 90 Chapter 4Elementary Survey Sampling, 7E, Scheaffer, Mendenhall, Ott and Gerow 22  a) Find sample size n.

EX , page 106 Chapter 4Elementary Survey Sampling, 7E, Scheaffer, Mendenhall, Ott and Gerow 23  a) Based on the information from EX4.27, find sample size n for EX4.28.

4.5 Estimation of a Population Proportion using SRS chosen w/o replacement in Finite Population (N)  Eg: Are you Hispanic or not? (YES / NO)  Define y i as 0 (if unit does not have quantity of interest) and y i =1 (if unit does have quantity of interest)  Then p-hat =  y i /n. (special case of sample mean)  p-hat is an unbiased estimator of p.  Estimated variance of p-hat (for infinite sample sizes) is p- hat*q-hat/n  Estimated variance of p-hat (for finite sample sizes) is (1- n/N)(p-hat*q-hat)/(n-1), where q-hat= 1-p-hat  Bound = 2*sqrt(Estimated variance of p-hat)  Problem 4.14

4.5 Estimation of a Population Proportion using SRS chosen w/o replacement in Finite Population (N)

EX4.14, page 104 Chapter 4Elementary Survey Sampling, 7E, Scheaffer, Mendenhall, Ott and Gerow 26

To estimate sample size  n = Npq/( (N-1)D + pq ) where D = B 2 /4  If p is unknown, then we use p = 0.5  Normality is important here!!  Problem 4.15  Question: All the bounds that we have looked at so far assumes what level of confidence?

4.6 Comparing two Estimates  Comparing two means, or two totals or two proportions:  Quantity of interest is  hat1 -  hat2  Variance of quantity of interest is V(  hat1 ) + V(  hat2 ) – 2cov(  hat1,  hat2 ) ********NOTE: We will NOT be using finite population correction factor in this section!!  If statistics come from two independent samples, then cov(  hat1,  hat2 ) = 0  Problem 4.18

4.6 Comparing two Estimates  When comparing means, we consider only the independent-sample case because the dependent case becomes too complicated to handle at this level.  When comparing proportions, however, a commonly occurring dependent situation does have a rather simple solution. Chapter 4Elementary Survey Sampling, 7E, Scheaffer, Mendenhall, Ott and Gerow 29

4.6 Comparing two Estimates  When comparing means, we consider only the independent- sample case because the dependent case becomes too complicated to handle at this level.  When comparing proportions, however, a commonly occurring dependent situation does have a rather simple solution. Chapter 4Elementary Survey Sampling, 7E, Scheaffer, Mendenhall, Ott and Gerow 30

EX4.18, page 104: comparing means Chapter 4 31

Eg4.11, page 98 Chapter 4Elementary Survey Sampling, 7E, Scheaffer, Mendenhall, Ott and Gerow 32

Extra Examples  A question asked to high school students was if they lied to a teacher at least one during the past year. The information is presented below MaleFemale Lied at least once Yes No Find the estimated difference in proportion for those who lied at least once to the teacher during the past year by gender. Place a bound on this estimated difference.* *Source: Moore, McCabe and Craig

Extra Multinomial example  If statistics are from a multinomial distribution, then cov(  hat1,  hat2 ) = (-p 1 p 2 /n)  In a class with 30 students, the table below illustrates the breakdown of class: Freshmen10 Sophomore5 Junior7 Senior8 Estimate the difference in percent Freshmen and percent Junior and place a bound on this difference.