t-based Confidence Interval for the Mean

In practical applications, we replace the population
standard deviation in
( 7.2) by *S*, the standard deviation
*of the sample*. However, this substitution changes the coverage
probability .
Fortunately, there is a simple adjustment that allows us to maintain
the desired coverage level :
replace the normal distribution critical value *z*
by the slightly larger *t*-distribution critical value *t*.
The resulting confidence interval is the primary result of this section.

where

The value *n*-1 is called
*degrees of freedom* , or *df* for short.
It is a parameter of the *t*-curve in the sense that changing the value of
*n*-1 changes the shape of the *t*-curve, though usually not by much.
Here are appropriate *t* critical values for selected
and *n*-1.

The *t* critical values are always larger than the *z*,
and get progressively closer
as *n*-1 gets larger (they are equal at ).
For a 95% confidence interval, the *t*
values are 2.06, 2.03, 2.01, 1.98, and 1.96 for
respective sample sizes *n*= 26,36, 51, 101, and 501.

Recall that the term
in equation ( 7.5)
is the (estimated) standard error of the mean. With .68 chance,
misses
by less than this amount.
To generalize,
misses
by less than
with
certainty. Thus,
the term
is called the *margin of error* with confidence level
.

If
,
then *t* is close to 2.0.
For this reason, the 95% margin of error is often written as
.
When working with a random sample, the exact critical
value *t* is read from a table or calculator, and depends on the sample size.
However, for sample size calculations (see next section), the
approximate critical value 2.0 is typically used.

**Example:**Given the following GPA for 6 students: 2.80, 3.20, 3.75, 3.10, 2.95, 3.40

a. Calculate a 95% confidence interval for the population mean GPA.

b. If the confidence level is increased from 95% to 99% , will the length of the confidence interval increase, decrease, or remain the same?

c. If the confidence level is kept at 95% but the sample size is quadrupled to*n*=24

(i) do you expect the sample mean to increase, decrease, or remain approximately the same?

(ii) do you expect the sample SD*S*to increase, decrease, or remain approximately the same?

(iii) do you expect the length of the confidence interval to increase, decrease, or remain approximately the same?

We end this section with a reminder that the
confidence interval ( 7.5)
works well ONLY when at least one of the following
conditions hold: (i) the population histogram looks
like the normal curve, or (ii) the sample size is relatively large
(
is a frequently used thumb rule .
The following table summarizes the performance of the *t*-intervals
under four situations.

Normal curve | Not normal curve | |

Small sample size [n<30] |
Good | Poor |

Larger sample sizes [] | Good | Fair |