handout_7_1 - 7.1 Inference for the Mean of a Population...

7.1 Inference for the Mean of a Population
the sampling distribution of x depends on σ
when σ is unknown, we must estimate σ even though we are primarily interested in μ
the sample standard deviation (s) is used to estimate the population standard deviation (σ)
x
has
N(μ,
n
σ
) distribution when population has N(μ, σ )
when σ is unknown, we estimate it with the sample standard deviation (s)
we estimate the standard deviation of
x
by
n
s
Standard error
When the estimated standard deviation is estimated from the data, the result is called the
standard error of the statistic. The standard error of the sample mean is
x
SE
=
n
s
one-sample z statistic (6.2) = z =








-
n
x
σ
μ ) (
0
basis for inference about μ when σ is known
x distributed normally (or approximately normally)- used Table A
when we substitute
n
s
for
n
σ
, our statistic is not distributed normally
the statistic now has a t distribution
The t distributions
Suppose that an SRS of size n is drawn from a N(μ, σ) population. Then the one-sample t
statistic
t =








-
n
s
x ) (
0
μ
has the t distribution with n-1 degrees of freedom .

•there is a different t distribution for each sample size (Table D)
•a particular t distribution is specified by giving the degrees of freedom
•we use t(k) to stand for the t distribution with k degrees of freedom
•The density curves of the t(k) distributions are similar in shape to the standard normal
curve (unimodal, symmetric about 0, and bell-shaped).