8.2. Testing a Proportion Hypothesis http://www.ck12.org
Since our critical value is 1.64 and our test statistic is 1.51, wecannot reject the null hypothesis. This means
that we cannot conclude that the population proportion is greater than 0.70 with 95 percent certainty. Given this
information, it is not safe to conclude that at least 70 percent of the voters would favor this bill with any degree of
certainty. Even though the proportion of voters supporting the bill is over 70 percent, this could be due to chance
and is not statistically significant.
Example:
Admission staff from a local university is conducting a survey to determine the proportion of incoming freshman
that will need financial aid. A survey on housing needs, financial aid and academic interests is collected from 400 of
the incoming freshman. Staff hypothesized that 30 percent of freshman will need financial aid and the sample from
the survey indicated that 101( 25 .3%)would need financial aid. Is this an accurate guess?
Solution:
First, we develop our null and alternative hypotheses.
H 0 :P= 0. 30
Ha:P 6 = 0. 30Next, we should set the criterion for rejecting the null hypothesis. The 0.05 alpha level is used and for an ?= 0. 05
the critical values of the test statistic are 1.96 standard deviations above or below the mean.
To calculate the test statistic, we first find the standard error of the proportion.
Sp=√
PQ
n=
√
0. 30 × 0. 70
400
≈ 0. 0229
After finding the standard error, we can calculate the standardz-score needed to evaluate our hypothesis.
Z=
p−P
sp=
0. 25 − 0. 30
0. 0229
≈− 2. 18
Since our critical value is 1.96 and our test statistic is− 2 .18, wecan reject the null hypothesis. This means that
we can conclude that the population of freshman needing financial aid is significantly more or less than 30 percent.
Since the test statistic is negative, we can conclude with 95% certainty that in the population of incoming freshman,
less than 30 percent of the students will need financial aid.
Confidence Intervals for Hypotheses about Population Proportions
When making a decision, we like to be able to determine how confident we are about a decision. For example, when
a congressman is deciding whether or not to vote for a bill, he would like to be able to say something to the effect of
“I am 99% confident that 70 percent of my constituents will support this decision.” With statistical analysis, we can
construct something called theconfidence intervalthat specifies the level of confidence that we have in our results.
The confidence interval is a range of values that we are confident, but not certain, contains the population parameter
that we are studying (most often this parameter is the mean).
We interpret the results of the confidence intervals by calculating:
- The level of confidence (i.e.−95%,99%,etc.)