8.2 & 8.3
Choosing a sample size
  What
sample size n do we need for a
given level of confidence about our
estimate.
 Larger
n coincides with better estimates and
more confidence (all else being equal).
  Sample
size calculations can be done for
estimating both µ and p.
1
Can we choose a level of
confidence AND have a certain
Margin of Error (MOE)?
  If
a client wants a smaller MOE for her 95%
confidence interval (i.e. a smaller range or
window for which we think the parameter
lies), she’ll have to take a larger sample.
  “We
don’t get something for nothing”.
2
Sample Size for Population Mean µ.
  Recall:
95% Confidence Interval (CI) for a
Population Mean µ
  The
margin of error (MOE) for the 95% CI
for µ is
2s
MOE = E ≈
n
where s is the standard deviation of the
sample. And the 95% CI is: X ± MOE
3
  Example
1:
A manufacturer of cereal boxes wants to
know the mean weight of the boxes it
produces. Previous studies have shown the
population standard deviation of the weights
of the boxes to be 0.1 ounces (i.e., σ = 0.1 ).
They would like to estimate µ with 95%
confidence AND have the MOE no greater
than 0.012.
4
  The
95% CI for µ depends on the MOE
which depends on s and the sample size n.
2s
MOE = E ≈
n
  The sample standard deviation s is an
estimate for the population standard
deviation σ. If we happen to know σ,
we will use it instead of s for our MOE.
2σ
MOE = E ≈
n
5
  This
client has asked that the MOE be no
greater than 0.012, and we know σ = 0.1 .
2σ
MOE =
n
2(0.1)
0.012 =
n
  Solving
for n gives…
6
2(0.1)
0.012 =
n
2(0.1) 2
2
(0.012) = (
)
n
0.04
0.000144 =
n
n = 277.78
To have a 95%
confidence interval
for µ with a MOE of
0.012, this
company will have
to sample 278
boxes.
But we can’t sample a ‘part of a box’,
so we need to round-up to n=278.
7
What Sample Size?
  The
sample size needed to be 95% confident
that X , the sample mean, will be within MOE
of the population mean, µ.
" 2σ %
n ≥$
'
# MOE &
2
8
Sample Size for Population
Proportion p.
  Recall:
95% Confidence Interval (CI) for a
Population Proportion p
  The
margin of error (MOE) for the 95% CI
for p is
ˆ
ˆ
p(1−
p)
MOE = E ≈ 2
n
where pˆ is the sample proportion. And
the 95% CI is: pˆ ± MOE
9
  Example
1:
A manufacturer of coats wants to know the
proportion of coats, p, it is producing with
defective zippers. They would like to
estimate p with 95% confidence AND have
the MOE no greater than 0.04.
10
  At
95% confidence, the MOE depends on
pˆ and the sample size n.
ˆ
ˆ
p(1−
p)
MOE = 2
n
Problem: we
ˆ
don’t know p
It turns out that the margin of error is largest
ˆ = 0.5. So, since we don’t know pˆ
when p
before we collect our data, we’ll plug-in p
ˆ =
0.5 which gives us the widest possible MOE
(i.e. we’re assuming worst case scenario for
estimating p).
11
plugging-in pˆ =0.5, and the
requested MOE from the client, this gives
us 1 equation with 1 unknown:
  After
ˆ
ˆ
p(1−
p)
MOE = 2
n
Replace each
pˆ with 0.5
0.5(1 − 0.5)
0.04 = 2
n
Now we can solve for n, the sample
size needed for this specific MOE.
12
0.5(1 − 0.5)
0.04 = 2
n
0.5(0.5) 2
2
(0.04) = (2
)
n
0.5(0.5)
0.0016 = 4
n
0.5(0.5)
n=4
0.0016
1
n=
= 625
0.0016
To have a 95%
confidence interval
for p with a MOE of
0.04, this company
will have to sample
625 coats.
13
What Sample Size?
  The
sample size needed to be 95% confident
ˆ , the sample proportion, will be within
that p
MOE of the population proportion, p.
1
n≥
2
MOE
14
Example 2:
  Suppose
we want to be 95% confident that
our sample proportion will be within 0.02 of
the population proportion.
1
1
n≥
⇒n≥
= 2, 500
2
2
MOE
(0.02)
Then we need to sample 2,500 people.
15
Example 3:
  The
Gallop Poll often quotes a margin of
error of +/- 3% at a confidence of 95%.
 How
many people did they ask*?
1
1
1
n≥
=
=
= 1111.11
2
2
MOE
(0.03)
0.0009
But we can’t sample a ‘part of a person’, so
we need to round-up for n=1112.
16
Example 4:
  USA today
  http://www.usatoday.com/news/polls/tables/live/
2007-02-12-poll.htm
 
“Results are based on telephone interviews with 1,006
National Adults, aged 18+, conducted February 9-11,
2007. For results based on the total sample of National
Adults, one can say with 95% confidence that the margin
of sampling error is ±3 percentage points.”
 
Technically, a MOE of 0.03154 coincides with a sample of size
n=1006 based on the formulas presented here.
17