http://www.ck12.org Chapter 8. Hypothesis Testing
- The interval (i.e.− 40 .4 to 45.6 or 102 to 108, etc.)
If we are estimating the confidence interval for a population mean, then we use the sample mean for the statistic.
However, if we are estimating for a population proportion, we use the sample population proportion.
The confidence interval always includes the population parameter. Therefore, when we construct a confidence
interval we can conclude that that interval also contains the sample statistic. A confidence interval statement would
look something like:
- We are 95 percent confident that the interval from 34.2 to 39.1 contains the mean
- ( 2. 10 ,<μ< 2. 90 )– We are 90 percent confident that this interval contains the population proportion
We cannotsay that the probability is 95 percent that the interval contains the mean since either the interval contains
the mean or it does not. Therefore, when we talk of our confidence level we say that we ’X% certain” that the
specific interval contains the mean.
Example:
In our example about the congressman voting for the bill on gay marriage, the congressman decides that he wants an
estimate of the proportion of voters in the population that are likely to vote for a bill. Construct a confidence interval
for this population proportion.
Solution:
As a reminder, our sample proportion was 0.746 and our standard error of the proportion was 0. 0265 .To correspond
with the ?=. 05 ,we will construct a 95% confidence interval for the population proportion. Under the normal curve,
95% of the area is betweenz=− 1 .96 andz= + 1. 96 .The confidence interval for this proportion would be:
CI 95 :
p± 1 .96(standard error)
0. 746 ±( 1. 96 )( 0. 0265 )So 0. 694 <p< 0. 798
With respect to the population proportion, we are 95% confident that the interval from 0.69 to.077 contains the
population proportion. This means that we are 95% confident that the average proportion of voters who will support
the bill is between 69 and 77%.
Lesson Summary
- In statistics, we also make inferences about proportions of a population. We use the same process as in testing
hypotheses about populations but we must include hypotheses about proportions and the proportions of the sample
in the analysis. - To calculate the test statistic needed to evaluate the population proportion hypothesis, we must also calculate the
standard error of the proportion which is defined assp=
√
PQ
n- The formula for calculating the test statistic for a population proportion is
z=
p−P
sp