__One Sample Proportion Test__

**T**o test the proportion of one
certain outcome in a population which follows **Bernoulli distribution**,
we can use the **1_proportion** function in **Minitab**.
This function produces a confidence interval and hypothesis test
of the proportion. The data format can be either raw( in the form
of "failure" and " success") or summarized.
Our example here is based on summarized data. Since we get summarized
data most of the time for this kind of experiments. If it is not
the case, **Tally **command (see Generate Random Data) can give us summarized
data right away.**I**n
Exercise 8.3 (BPS, Chapter 8, page 431), the college president
says "99% of the alumni support my firing of coach Boggs."
However in the SRS of 200 out of the colleges 15,000 living alumni
only 76 of them support firing coach. Now we need procedure a
hypothesis test to address the question whether the proportion
of alumni who support firing coach is equal to 99% as the president
said. Hence, we are going to test: Ho:
*p*=0.99 Ha: *p*=/=0.99

Select **Stat->Basic Statistics->1
proportion** from the menu. Since we have only summarized data
in this case, which is 200 trails and 76 successes, check the
box "**Summarized data**", fill out the number in
appropriate location.

The
**Option** subdialog allows us to specify the confidence level,
test probability, alternative hypothesis, and whether **Minitab**
should employ test statistic and interval based on **Normal distribution**.
If this box is checked, **Minitab** is going to calculation
C.I. as , and the test statistics
(**z**) as. Otherwise, **Minitab**
will use an exact method to calculate the test probability by
default. There are slight difference between the two method, which
could become significant when n is small.

The 95% C.I. of the proportion who support firing coach is (0.313, 0.447), and the P-value shown in output is 0.000 which gives us strong evidence to reject null hypothesis.

With two binomial proportions in our hand, one frequently asked question is whether they are equal or not. In the word of statistics, the following hypothesis needs to be tested:

In Exercise 8.24 (BPS Chapter 8, page 451), 161 people who visited one hospital's emergency room in a 6-month study period with injuries from in-line skating were interviewed. The interviewer found that 53 people were wearing wrist guards and 6 of them had wrist injuries. Of the 108 who did not wear wrist guards, 45 had wrist injuries. We are interested in the difference between the proportions of wrist injuries in the population wearing wrist guard and the population without.

Select **Stat->Basic
Statistics->2 proportions** from menu, since we got summarized
data here, check the box of **"Summarized Data"**
and fill it out as shown.

The **Option** subdialog box gives
us a chance to specify the confidence level, test proportion,
alternative hypothesis, and whether Minitab should use a pooled
estimate of *p* for the test.

This **2-proportions** function calculates
the confidence interval as :

By default, **Minitab **uses separate
estimates of p for each population and calculates **z** as
:

Once we check the box of "**Use
pooled estimate of p for test**", **Minitab** calculates
**z** as :, with .

The
output shows that the **P-value** is 0.000, so there is strong
evidence that wearing wrist guard reduce the rate of wrist injuries.