Stat 4473 – Data Analysis
Sample size calculations
All the behaviors of confidence intervals discussed earlier still apply.
(1) higher confidence levels ] larger margins of error
(2) larger samples
] smaller margins of error
(3) populations have high variability ] larger margins of error
There’s nothing we can do about (3), but if we demand a certain high level of confidence
and a certain precision in our interval estimate, we can choose a sufficiently large sample
size. Let E = specified margin of error.
One sample t-interval for :
But, we haven’t taken the sample yet, so we can’t calculate the sample standard deviation,
s
, nor t* since the df depends on
n
.
•For
s
, make a good guess or run a small pilot study.
•For
t*
, the common approach is to use the corresponding
z*
value.
•Note
: Sample size calculations are never exact. The margin of error you find after
collecting the data won’t match exactly the one you specified to find
n
. The sample
size formula depends on quantities that you won’t have until you collect the data,
but using it is an important first step.
Example
An economist wants to estimate the mean income for the first year of work for college
graduates who have taken a statistics course. He requires 95% confidence that the
sample mean is within $500 of the true population mean. How large should his sample be?
Solution: The specified margin of error is $500. So, the sample size should be chosen
so that . Suppose a pilot study involving 30 individuals has a
standard deviation of $6250. We can’t calculate
t*
, so we’ll use the
corresponding
z*
value, which is
z
.975 = 1.96 for 95% confidence.
Then, , so the economist should sample 601
individuals. (He can use the 30 he already has and get 571 more.)
