8.2. Population Mean: Small Sample Case

t-distribution (“student” t distribution):

This distribution was invented by W. S. Gossett (published in the name “Student”).

The t-distribution is a family of similar probability distributions, with a specific t distribution

depending on a parameter known the degree of freedom (d.f.). Denote to be the

random variable having t-distribution with n degree of freedom.

Note: . The t-distribution is symmetric about 0.

: a t value with an area of in the upper tail of the t-distribution with degree of

freedom equal to n.

.

Important Result:

When the population has a normal probability distribution and is

unknown, then

,

where , , and are random

variables with associated possible values .

Derivation of confidence interval:

Suppose the population has a normal probability distribution. Since

thus

confidence interval based on t-distribution:

As the population has a normal distribution and is unknown,

is a confidence interval of .

Example 2:

Consider the following random sample of 4 observations, 25, 47, 32, and 56. Suppose

the population is normally distributed. Please construct a 95% confidence interval for .

[solution:]

In addition, and . Thus,

JavaStatSoft:

Confidence interval:

Statistics -> Estimation -> One Sample -> Mean -> T Interval