Math 207—More
Confidence Interval Examples
Example 1
(Confidence interval for a population proportion)
For marketing
purposes, a magazine is interested in the proportion of males in its
readership. The marketing director takes a random sample of 70 readers (how
might this be done? It wouldn’t be easy.), and finds that 62% of them are male.
Find and interpret a 99% confidence interval for the proportion of males in the
readership population.
Example 2 (Confidence interval
for a difference in population means)
A researcher is
interested in the relationship between cocaine use during pregnancy and the
birth weight of the baby. Birth weights (in grams) of babies of women who
tested positive for cocaine were compared with the birth weights (in grams) for
women who tested negative. The sample data are shown in the table below.
|
Group |
Sample
size, n |
Sample
mean, |
Sample
standard deviation, s |
|
Positive test |
134 |
2733 |
599 |
|
Negative test |
5974 |
3118 |
672 |
Find a 95% confidence interval for the
difference in population mean birth weights.
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Note it doesn’t matter
which way you do the differencing, as long as you make the appropriate
interpretation.
Method 1:
95%
confidence interval for 
is
We
are 95% confident that the difference in mean baby birth weights (drug-using
population – non-drug-using population) is between -487.9 grams and -282.2
grams. Thus, we are 95% confident that babies born to cocaine-using moms weigh
less on average than babies born to non-cocaine-using moms (since the interval
is completely negative). As always, our confidence is in the method we use—the
method is correct 95% of the time.
Method 2:
95%
confidence interval for 
is
We
are 95% confident that the difference in mean baby birth weights
(non-drug-using population –drug-using population) is between 282.2 grams and
487.8 grams. Thus, we are 95% confident that babies born to non-cocaine-using
moms weigh more on average than babies born to cocaine-using moms (since the
interval is completely positive). As always, our confidence is in the method we
use—the method is correct 95% of the time.
Note that both methods
give the same conclusion.
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Can we say that cocaine uses causes low birth weight? Why or why
not?
Example 3 (Confidence interval
for a difference in population proportions)
Managers at two
different factories (call them A and B) want to compare the proportion of
defective items produced by their factories. They each take a random sample of
items and find the following proportion of defectives:
|
Factory |
Sample
size, n |
Sample
proportion, |
|
A |
500 |
0.032 |
|
B |
600 |
0.045 |
Find and
interpret a 98% confidence interval for the difference in defective proportions
for the two populations.
Example 4 (Sample Size Determination)
Suppose the
Appleton Chamber of Commerce is interested in the average family income of all
Appleton households. They want to estimate the mean with 95% confidence, yet
they want the margin of error to be no more than $2500. A small pilot study was
done and from the sample, s = $23,130.
How large of a sample should they take?
Example 5 (Sample Size
Determination)
The Chamber of
Commerce is also interested in the proportion of
How large of a
sample should they take?
Important note: Examples 4 and 5 illustrate sample size
determination for one-sample
problems. We can also determine sample sizes for two-sample problems. Two things change for the two-sample
problems: 1) You must use the appropriate standard error for the two-sample
confidence interval, and 2) you must assume equal sample sizes—that is, 
.