**ONE SAMPLE** **T-TESTS
**

**ONE SAMPLE T-TEST OF MEANS: **Univariate
Inference for Interval/Ratio Data

A. Large samples with a known population mean.

Most of the time, the population's standard deviation is not available, so it is substituted by the sample's standard deviation .

**where: N = total number in the sample.**

**= mean of the sample**

**= population mean **

**= population standard deviation**

**S = sample standard deviation
**

**Assumptions: N>100**

**Population mean known**

**Example: **Anecdotal evidence
indicates that students in 2:00 classes work less hours in the
labor force than other students. The average student at the college
works 23.8 hours a week. A sample of 147 students in 2:00 classes
has a mean average of 22.47 hours a week and a standard deviation
of 8.15 hours.

**Q:** Is the difference shown
in the sample generalizeable to the whole population of students
at the college?

**A: Since there is
a direction to the hypothesis, "less hours" indicates
a negative relationship, the one-tail Z-score is used. **

**TWO TAILED TEST ONE TAILED TEST**

**Z p=.05 p=.01 p=.05 p=.01**

**Z 1.960 2.576 1.645 2.326**

The probability of being wrong if when generalizing
the difference in average work hours is less than 5%. Therefore,
the anecdotal evidence is supported by this sample and, in general,
students in 2:00 classes work less hours in the labor force than
other students.

B. Small Samples with population means known.

**t**=
, **d.f. = N-1
**

**Where**: **=
mean of the sample**

**= population mean **

**S = sample standard deviation**

**N = total number in the sample**

**d.f. = degrees of freedom
**

**Assumption: Population mean known.**

**Example: **Using the same
data as above, except the sample size is now only 17, is the difference
still significant?

**t**=
, **d.f. = 17-1=16
**

**Look at the significant values for t-test chart
for the one-tailed test..**

**-------P=.05 ---------P=.01**

**df**

**16----- 1.76 ----------2.583**

P>.05,** **this average falls within the range
that 95% of the means fall, the null hypothesis is NOT rejected.
The difference in means is not significant and cannot be generalized
to the population of students in 2:00 classes.

**ONE SAMPLE T-TEST OF PROPORTIONS:**
Univariate inference for nominal data.

In many cases, analyst do not have interval/ratio level data. The following is an inferential test based on the same logic as the t-test of means, but using sample proportions and a known population proportion.

**where: P_{s }= Sample proportion**

**P_{u }= Population proportion**

**N = Sample total
**

**Assumptions: N>100**

**Population proportion known.
**

**Example: **In Community
A, 10.9% of the people know someone with AIDS. Taking a random
sample of 104 people at the senior citizen center, only 5% of
the sample knows someone with AIDS.** **

**Q: **Is this difference
significant?

A:

Look on the Z-chart, P<.05, therefore, the null
hypothesis that there is no relationship between being a senior
citizen and knowing someone with AIDS is rejected. Our conclusion
is that fewer senior citizens people in Community A know people
with AIDS than the rest of the population.

**TWO-SAMPLE T-TEST OF MEANS: **Bivariate
inferential for interval/ratio data

In this case, the population descriptive statistics
are not necessary.

A. For Large Samples:

**Where: = the mean of
each sample.**

**S_{i }= the standard deviation of each
sample**

**N_{i }= sample total of each sample.
**

**Assumptions: N > 100**

**Independent samples ***(this
can be difficult to prove, assume independence for this class)
*

**Example: In **random samples
of 143 female and 112 male SDSU students, the average hourly wage
for female students is $6.73 with a standard deviation of $2.58.
For the male students, it is $7.72 with a standard deviation of
$4.79.

**Q:** Is this difference
in hourly wages between male and female students generalizeable
to the whole population of SDSU?

There is a 5% chance of being wrong if we state that
gender is associated with hourly wages of SDSU students.

B. For Small Samples:

**d.f = N _{1 }+
N_{2 }- 2.**

**Where: = the mean of
each sample.**

**S_{i }= the standard deviation of each
sample**

**N_{i }= sample total of each sample.
**

**d.f. = degrees of freedom.**

**Assumptions: Independent samples**
*(this can be difficult to prove, assume
independence for this class)*

**Similar Population variances**
*(Similar variance can be proven by Bartlet's
test, which is beyond the scope of this class, so assume similar
variance as well)*

**Example: **This time find
out if there is a relationship between gender and hours spent
working in the labor force within SDSU's student population. The
sample average of 23 male students is 25 hours with a standard
deviation of 9.27 hours. The sample average of 30 female students
is 22.71 hours with a standard deviation of 8.15 hours.

**A:**
**d.f = 23 _{ }+ 30_{ }-
2 = 51**

Look in the back of the book for the critical values
of the t-test with the a d.f. of 30.

**d.f.----P=.05 --------P=.01**

**30 ------2.042-------- 2.75 for two-tailed **

**30 ------1.697 --------2.457 for one-tailed
**

Since the hypothesis did not indicate a direction (i.e. stating being a female negatively relates to hours worked in the labor force as compared to being a male), use the two-tailed critical values and fail to reject the null hypothesis that there is no relationship. Therefore, do not generalize. If the hypothesis had indicated a direction, using the one-tailed test would have given a probability value of less than 5% and the above difference in means could be generalized to the whole SDSU student population.

**TWO-SAMPLE T-TEST OF PROPORTIONS:**
Bivariate inferential statistic for nominal data.

**where:**

**= the mean of each sample.**

**P_{i }= the proportion of each sample**

**N_{i }= sample total of each sample.
**

**Assumptions: N > 100**

**Example: 17.4% of 115 male and 27.3% of 132 females
SDSU students randomly surveyed answered "yes" to the
question "Do you know anyone who has contracted the HIV virus?"
**

Q:** **Can you generalize the difference to the
whole SDSU student population?

**A: **First, estimate the
population proportion.**
**

Then calculate the standardized score.**
**

P < .01, There is less than a one percent chance that there is no relationship between gender and knowing someone who has contracted the HIV virus among the SDSU student population. So yes, this sample's gender difference can be generalized to the whole population.