Transcript 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.

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.
pˆ 1  pˆ 
pˆ  z *
n
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.
p * 1  p *
m  z*
n
• 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!
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*(1-p*)
.25
p*=0
.5
1
p*
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!
.7(.3)
.7  1.96
20
.7  .2008
(.4992,.9008)
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
.03  1.96
.5(.5)
n
Divide both sides by 1.96
.0153
.25
n
Square both sides
• Solve for n
.000234
.25
n
Multiply both sides by n
.000234n  .25
• Round up!
Divide both sides by.000234
n  1068.376
n  1069
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.