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.