Simple Random Sampling, Study notes of Accounting

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Simple Random Sampling!

Professor Ron Fricker! Naval Postgraduate School! Monterey, California! 2/1/13 1 Reading Assignment:! Scheaffer, Mendenhall, Ott, & Gerow! Chapter 4!

Goals for this Lecture!

• Define simple random sampling (SRS) and discuss how to draw one!
• Horvitz-Thompson estimation and SRS!
  • The finite population correction (fpc)!
• Defining estimators for means, totals, and proportions!
• Sample size calculations!

Example!

• Consider a population consisting of 90 men and 10 women, so N =100, where we want to sample n =10 individuals! - With SRS, we can get samples of all men or all women!
• We could also draw a stratified sample, where via SRS we sample nine men and (separately) via SRS one woman! - Here each person has probability 1/10 of being sampled, but not all groups of 10 can be sampled!

How to Draw a SRS!

• Easiest way:!
  • Assign every element in the sampling frame a uniformly distributed random number (say between 0 and 1)!
  • Sort the list according to the random numbers!
    • Either ascending or descending, doesn’t matter!
  • Then take the first n elements!
• Don’t try to actually generate all possible combinations of n elements out of N …!
• Chapter 4 describes other manual ways to do this using tables of random numbers!

Note the Difference!

• So, notice that giving every element in the population an equal chance of selection like this results in a SRS!
• Which is probably why SRS is often mistakenly defined this way!
• But remember that other non-SRS methods can also result in every element having an equal chance of being selected! - For example, stratified sampling when probability of selection is proportional to strata size!

Horvitz-Thompson Under SRS!

• Under SRS, each sampling unit has probability n / N of being selected!
• Estimating with Horvitz-Thompson estimator, we have! - Same as Stats 101!!
• If population is infinite, standard error of is estimated the same way too:! 1 1 1 1 1 1 1 1 1 1 ˆ / n n n n i i i i i (^) i i i i N y y y y y N N n N N n n μ = π = = = = = = = = ∑ ∑ ∑ ∑ ˆ y σ = s n

y

Finite Population Correction!

• Note that failure to use the finite population correction (fpc) results in standard errors that are too large! - Confidence intervals will be (erroneously) too big! - Hypothesis tests will be (erroneously) less powerful!
• For a survey with sample size less than 5 percent of population, can ignore the fpc! - It will have negligible effect!
• If sample size larger than 5 percent, use fpc to get more precise results – a good thing!!

Example: Margin of Error Estimates!

• For various sample sizes, margins of error for an infinite- sized population and one with N = - Binary question! - Conservative p =0.5 assumption!

Where Does the FPC Come From?!

• In an infinite population, if we sample two observations then! - Doesn’t really matter whether we sample with replacement or not!
• For a finite population, when we sample without replacement,!
• Picking one observation affects the rest, so there is correlation!! Cov( , ) 0 i j Y Y = (^12) Cov( , ) 1 i j Y Y N

−

Mean Estimation Summary!

• Estimator for the mean:!
• Variance of :!
• Bound on the error of estimation (margin of error):! 1 1 n i i y y n (^) = = ∑ y (^) Var 

( y ) =^1 −^

n N ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ s 2 n 2 Var 

( y ) =^2 1 −^

n N ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ s 2 n

Estimating Proportions!

• Estimator for the proportion:!
• Variance of :!
• Bound on the error of estimation (margin of error):! 1 1 ˆ n i i p y y n (^) = = = ∑ p^ ˆ (^) Var 

( p ˆ) =^1 −^

n N ⎛ ⎝ ⎜ ⎞ ⎠ ⎟

p^ ˆ ( 1 − p ˆ)

n 2 Var 

( p ˆ) =^2 1 −^

n N ⎛ ⎝ ⎜ ⎞ ⎠ ⎟

p^ ˆ ( 1 − p ˆ)

n

Sample Size Calculations (w/ fpc) for Estimating Means!

• Typically, we want to determine a sample size to achieve a particular margin of error B
• So, solving the following for n gives!
• This is the number of respondents required!
  • Will need to inflate to account for nonrespondents! 2 2 1 N n B N n ⎛ − ⎞ σ = ⎜ ⎟ ⎝ − ⎠ ( ) 2 2 2 1 4 N n B N σ σ = − +

Sample Size Calculations (w/ fpc) for Estimating Proportions!

• Again proceed as before, but use the expression for proportions!
• That is, solve the following for n gives!
• And again, don’t forget to inflate this to account for the nonresponse rate! 2 1 − n N ⎛ ⎝ ⎜ ⎞ ⎠ ⎟

p ( 1 − p )

n = B ( ) 2 (1 ) 1 4 (1 ) Np p n B N p p − = − + −

Power Calculations Example!

• Back to survey with N =300, where we guess that p =50% (most conservative assumption)!
• What sample size do we need to achieve a margin of error of 3%?!
• So, need responses from 237 out of the 300
  • If 80% response rate, must sample 237/0.8=297!! ( ) ( ) 2 2 (1 ) 1 4 (1 ) 300 0.5(1 0.5)

0.03 300 1 4 0.5(1 0.5) Np p n B N p p − = − + − × − = = − + −