**Example 1.** Suppose that an ad for Western Michigan University claims that
`a majority of WMU students come from outside the Southwest Michigan area'.
A sample of 25 students revealed that only 9 out of 25, or 36%,
come from outside Southwest Michigan. Can this just be chance error, or is this
significant evidence that the advertisement is wrong?

*Solution:*
The data consists of a *sample percentage* (or proportion); i.e. 36% of 25 students
come from outside Southwest Michigan. Therefore, the ESTIMATE is a either a sample
percentage or proportion (we will choose a proportion).
At the end of the test, the conclusion will be one of two things:
the data will provide evidence either for or against against the claim
`a majority of WMU students come from outside the Southwest Michigan area'.
Therefore the competing hypotheses are
versus *p* < .50.
The hypothesis with the equal sign is null, so we have
.
The *H*_{0} Value is .50.

- 1.
- Hypotheses:
versus
*H*_{1}:*p*< .50 - 2.
- Test Statistic:
- 3.
- P-value: Presuming
*H*_{0}is true, the likelihood of chance variation yielding a*z*-statistic as low as -1.40 is .08. - 4.
- Conclusion: Since P-value > .05, the observed sample value is not significantly different from .50. Hence, we do not reject . The sample does not provide enough evidence to prove the ad wrong.

**Example 2**
Is there "grade inflation" in WMU?
A random sample of 100 student records from 10 years ago yields a sample average
GPA of 2.90 with a standard deviation of .40. A random sample of 100
current students today yields a sample average
of 2.98 with a standard deviation of .45. Is the difference between
2.90 and 2.98 just chance variation, or a
statistically significant increase?

*Solution:*
The evidence consists of two sample averages. The ESTIMATE of interest
is the difference: 2.98-2.90=.08 (i.e average today minus average 10 years ago).
At the end of the test, the conclusion will be one of two things:
the data either support or provide evidence against the hypothesis
of 'grade inflation'. Therefore the competing hypotheses are
'average grade today is higher than 10 years ago' or 'not'. In terms of the
population averages, we write these as
,
or
.
The hypothesis with the equal sign is null, so we have
.
The *H*_{0} Value is 0.

- 1.
- Hypotheses: versus
- 2.
- Test Statistic:
- 3.
- P-value: Presuming
*H*_{0}is true, a*z*-statistic as large as 1.33 results from chance variation 9% of the time (P-value=.09). - 4.
- Conclusion: Since P-value > .05, the observed increase in average GPA is
not statistically significant, and may be explained by chance variation.
Do not reject
*H*_{0}.

If the sample sizes were small and the two variances were equal, a pooled-SD t-test
would have been the more appropriate test. We illustrate the procedure here.
The SD's for the two samples *S*_{1}=.40 and *S*_{2}=.45 are pooled into one SD:

The SE's for the two averages are and . Now we conduct the test.

- 1.
- Hypotheses: versus
- 2.
- Test Statistic:
- 3.
- P-value: Presuming
*H*_{0}is true, a*t*-statistic with 198 degrees of freedom will by chance be as large as 1.33 around 9% of the time (P-value=.09). - 4.
- Conclusion: Since P-value > .05, the observed increase in average GPA is
not statistically significant, and may be explained by chance variation.
Do not reject
*H*_{0}.

**Example 3. The paired t-test**
Table 8.2 shows stock prices for six selected food service companies.
Is there a significant difference between average Jan. 2002 and Jan. 2003 stock prices?

*Solution:*

At first glance, the evidence looks like it consists of two averages 31.12 and 28.07. However, remember that these two averages are not independent, because the data consists of two measurements on only ONE sample, rather than two separate samples. (Remember the Paired-t confidence interval?) The correct analysis here takes the differences between the two measurements, as shown on the table. If average 2002 and 2003 prices are significantly different, then the difference column will have an average that is significantly different from 0.

Ignoring the original two columns and focusing instead on the difference column, the evidence consists of one average. At the end of the test, the conclusion will be one of two things: the average difference is either significantly different from 0 or not ( versus ). Suppose we want a test that can detect either a positive or negative change in average stock prices, then we can choose to do a two-tailed test. (Of course, a one-tailed test that detects change in only one direction can also be done.)

- 1.
- Hypotheses: versus
- 2.
- Test Statistic:
- 3.
- P-value: By chance alone, a
*t*-statistic with 5 d.f. may get as large as 1.32 on either side of 0 around 24% of the time (P-value=.244). - 4.
- Conclusion: Since P-value > .05, the observed change in stock prices is
not statistically significant, and may be explained by chance variation.
Do not reject
*H*_{0}.

**Example 4.**
Has retention rate at WMU been changing? Suppose that a random sample of
200 entering students in 1989 showed 74% were still enrolled 3 years later.
Another random sample of 200 entering students in 1999 showed that 66%
were still enrolled 3 years later. Is this drop of 8% statistical evidence
of changing retention rate, or can it be just chance error (i.e
luck of the draw in the students selected for the samples)?

*Solution:*

The evidence consists of two sample proportions ,
or more precisely, the difference
between two sample proportions (.74-.66=.08). At the end of the test,
the conclusion will be one of two things:
the data either support or provide evidence against 'changing retention rates'.
Since an increase or decrease will be interpreted as a change (this was discussed
at the start of the study), we will conduct a two-tailed test.
The competing hypotheses are 'change' versus 'no change'.
In terms of the population, these may be written as
versus *p*_{1}-*p*_{2}=0. The latter is assigned to *H*_{0} because
it contains the equal sign. The *H*_{0} Value is 0.

- 1.
- Hypotheses:
*H*_{0}:*p*_{1}-*p*_{2}=0 versus - 2.
- Test Statistic:
- 3.
- P-value: Presuming
*H*_{0}is true, a*z*-statistic as extreme as 1.78 on either side of 0 results from chance variation 7.5% of the time (P-value=.075). - 4.
- Conclusion: Since P-value > .05, the observed change in retention rate is
not statistically significant, and may be explained by chance variation.
Do not reject
*H*_{0}.