AP Statistics 2017

of larger samples are less variable than sample means of small samples. However, it is often expensive
or difficult to draw larger samples so that we try to find the optimum sample size: large enough to
accomplish our goals, small enough that we can afford it or manage it. We will look at techniques in this
section for selecting sample sizes in the case of a large sample test for a single population mean and for a
single population proportion.

Sample Size for Estimating a Population Mean (Large Sample)

The large sample confidence interval for a population mean is given by The margin of error is

given by . Let M be the desired maximum margin of error. Then, Solving for n , we

have . Using this “recipe,” we can calculate the minimum n needed for a fixed confidence

level and a fixed maximum margin of error.
One obvious problem with using this expression as a way to figure n is that we will not know σ , so
we need to estimate it in some way. In an exam question, you will almost certainly be provided with an
estimate of σ to use in the calculation.
In reality, a researcher would probably be able to utilize some historical knowledge about the
standard deviation for the type of data they are examining, as shown in the following example:

example: A  machine for inflating   tires,  when    properly    calibrated, inflates    tires   to  32  lbs,    but it  is

known   that    the machine varies  with    a   standard    deviation   of  about   0.8 lbs.    How large   a   sample

is  needed  in  order   be  99% confident   that    the mean    inflation   pressure    is  within  a   margin  of  error

of  M = 0.10    lbs?

solution:

.   Since   n must  be  an  integer and n ≥ 424.49, choose

n = 425.    You would   need    a   sample  of  at  least   425 tires.

In this course you will not need to find a sample size for constructing a confidence interval involving t .
This is because you need to know the sample size before you can determine t since there is a different t
distribution for each different number of degrees of freedom. For example, for a 95% confidence interval
for the mean of a normal distribution, you know that z = 1.96 no matter what sample size you are dealing
with, but you can’t determine t * without already knowing n .

Sample Size for Estimating a Population Proportion

The confidence interval for a population proportion is given by:

The margin  of  error   is