Choosing Sample Size and Using Your Calculator Presentation 9.3
Choosing Sample Size and Using Your Calculator
Presentation 9.3
Margin of Error
• The margin of error (m) of a confidence interval is the plus and minus part of the confidence interval
• A confidence interval that has a margin of error of plus or minus 3 percentage points means that the margin of error m=.03.
n
ppzp
ˆ1ˆ*ˆ
Margin of Error
Margin of Error
• A common problem in statistics is to figure out what sample size will be needed to obtain a desired accuracy or margin of error.
• This is essentially algebra problem.
Determining Sample Size• Set up the following to obtain a margin of error m.• p* is you best guess of the proportion (remember you determine
sample size before you actually take the sample).– More on this p* later.
• Then, solve for n.
• Be sure to ALWAYS round up.– If you round, for example 5.023 to 5, your margin of error will come out just
a hair to big.– So, err on the side of caution and ALWAYS round up!
n
ppzm
*1**
Sample Size
• The margin of error desired m, is usually provided in the problem.
• The value z* is determined by the level of confidence that is desired (typically 90%, 95%, or 99%).
• The p* value is your best guess about the value of the true p.– So we are trying to do a study to estimate p, but we
need to know p or p* to compute the needed sample size. This seems impossible!
– What to do, what to do?
Sample Size
• Do the best you can.• Give the best or most current state of knowledge
about p as p*.• Many times there is some information or hint
about what p might be.• If you know absolutely nothing, then use p*=.5
as that will create the largest standard error and thus guarantee your margin of error.– This is again erring on the side of caution.
Why use p*=.5?• Here is a graph of p*(1-p*) for values of p*:
p*p*=0 .5 1
p*(1-p*)
.25
So you can see that using p*=.5 gives you the largest standard error.
Why use p*=.5
• The graph shows that p*(1-p*) will be largest when p*=.5.– This means the sample size will be largest when
p*=.5.– Which means that the sample size will be at least as
big as actually needed.
• This is being conservative as you are using more data than you would actually need to achieve the desired margin of error.
Sample Size Example #1: Home Court Advantage
• Home Court Advantage• In watching n=20 college
basketball games, it seems as if the home team usually wins.
• In fact, the home team won 14 times in 20 games.
• This means p-hat = 14/20 = .7 or 70% of the time!
• What is a 95% confidence interval for true home court win proportion p?
Sample Size Example #1: Home Court Advantage
• Calculate the confidence interval
• A 20% margin of error!
• That is unacceptable and a rather useless confidence interval!– It’s simply way too wide!
)9008,.4992(.
2008.7.20
)3(.7.96.17.
Sample Size Example #1: Home Court Advantage
• How big of a sample would we need?• How accurate (narrow interval or small margin of
error) would we like to be?• Suppose we wish to obtain a margin of error of
3% in a 95% CI for p.– That is, we want a proportion plus or minus 3%.
• How many games would I have to or get to watch?
Sample Size Example #1: Home Court Advantage
• Set up the equation– We need to guess p*– To be conservative,
use .5
• Solve for n
• Round up!
1069
376.1068
25.000234.
25.000234.
25.0153.
)5(.5.96.103.
n
n
n
n
n
nDivide both sides by 1.96
Square both sides
Multiply both sides by n
Divide both sides by.000234
Sample Size Example #1: Home Court Advantage
• Very cool!• I now have a
statistical reason for watching 1069 college basketball games!
Choosing Sample Size and Using Your Calculator
This concludes this presentation.