**Inference For the Mean of a
Population:**

If our data comes from a
simple random sample (SRS) and the sample size is sufficiently large, then we
know that the sampling distribution of the sample means is approximately normal
with mean _{} and standard deviation _{}.

**PROBLEM:**

If _{} is unknown, then we
cannot calculate the standard deviation for the sampling model. L

We must estimate the value
of _{} in order to use the
methods of inference that we have learned.

**SOLUTION:**

We will use *s* (the standard deviation of the sample)
to estimate _{}.

Then the **standard error** of the sample mean _{} is _{}.

In order to standardize _{}, we subtract its mean and divide by its standard
deviation.

_{} has
the normal distribution N( 0, 1)

**PROBLEM:**

If we replace _{} with *s*, then the statistic has more variation
and no longer has a normal distribution so we cannot call it *z*.

It has a new distribution
called the ** t distribution**.

_{}

t is a standardized value. Like *z*,
*t* tells us how many standardized
units _{} is from the mean _{}.

When we describe a t
distribution we must identify its degrees of freedom because there is a
different t statistic for each sample size.

The
degrees of freedom for the one-sample t statistic is
(n – 1).

The t distribution is
symmetric about zero and is bell-shaped, but there is more variation so the
spread is greater.

As the degrees of freedom
increase, the t distribution gets closer to the normal distribution, since s
gets closer to _{}.

We can construct a
confidence interval using the t
distribution in the same way we constructed confidence intervals for the z
distribution.

_{}

Remember, the t Table uses
the area to the **RIGHT** of t*.

One-sample t procedures are
exactly correct only when the population is normal. We assume that the population is
approximately normal in order to justify the use of t procedures.

A confidence interval or
significance test is called **robust** if
the confidence level or P-value does not change very much when the assumptions
are violated.

The t procedures are
strongly influenced by outliers. Always
check the data first!

If there are outliers and
the sample size is small, the results will not be reliable.

The t procedures are robust
when there are no outliers, especially when the distribution is approximately
symmetric.

**When to use t procedures:**

▪
If the sample
size is less than 15, only use t procedures if the data are close to normal.

▪
If the sample
size is at least 15, only use t procedures if there are no outliers.

▪
If the sample
size is at least 40, you may use t procedures, even if the data is skewed

**Comparing Two Means:**

Comparative studies are more
convincing than single single-sample investigations, so one-sample inference is
not as common as comparative (two-sample) inference.

In a comparative study, we
may want to compare two treatments, or we may want to compare two
populations. In either case, the samples
must be chosen randomly and independently in order to perform statistical
inference.

Because matched pairs are
NOT chosen independently, we will NOT use two-sample inference for a matched
pairs design. For a matched pairs
design, apply the one-sample t procedures to the observed differences.

Otherwise, we may use two-sample inference to
compare two treatments or two populations.

The null hypothesis is that
there is no difference between the two parameters.

_{} or _{}

The alternative hypothesis
could be that

_{} (two-sided)

_{} or _{} (one-sided)

_{} or _{} (one-sided)

Before you begin, check your
assumptions! For comparing two means,
both samples must be an **SRS****independently**. Also, both **populations must be normally distributed**. (Check the data for outliers or skewedness.)

If these assumptions hold,
then the difference in sample means is an unbiased estimator of the difference
in population means, so _{} is equal to _{}.

Also, the variance of _{} is the sum of the
variances of _{} and _{}, which is _{}.

Furthermore, if both
populations are normally distributed, then _{} is also normally
distributed.

In order to standardize _{}, subtract the mean and divide by the standard deviation:

_{}

If we do not know _{} and _{}, we will substitute the standard error _{} and _{} for the standard
deviation. This gives the standardized t
value:

_{}

**CAUTION!** This statistic does **NOT** have a t distribution.

We will use the TI-83 to
perform two-sample T tests.

**2-SampTTest**
for a hypothesis test

**2-SampTInt**
for a confidence interval