Download Inference for the Mean of a Population

Inference for the Mean of a Population
Inference for µ
The goal is to make inference about
x
is a point estimate of
µ
µ
.
If we sample from a normal distribution, the sampling
x−µ
distribution of
is standard normal.
Z=
σ
n
If we sample from a non-normal distribution, the sampling
distribution above is approximately standard normal for large n.
Standard Error
σ
is seldom known
Estimate σ by S
When the standard deviation of a statistic is estimated
from the data, the result is called the standard error of a
statistic. The standard error of the sample mean is
s
SE x =
n
The t Distributions
Suppose that an SRS of size n is drawn from a
N (µ , σ ) population. Then the one-sample t statistic
x−µ
t=
s
n
has the t distribution with n – 1 degrees of freedom.
The assumption that we sample from a normal population is
important for small n, but not for large n.
Properties of the t Distribution
continuous and symmetric about 0
more variable and slightly different shape than standard
normal (see Figure page 494)
There are an infinite number of t distributions. They are
indexed by the number of degrees of freedom.
As the number of degrees of freedom increase, the
t distribution approaches the standard normal.
The One-Sample t Confidence Interval
Suppose that an SRS of size n is drawn from a
population having unknown mean µ . A level C
confidence interval for µ is
S
x ±t*
n
• Where t* is the value for t(n – 1) density curve with
area C between –t* and t*. This interval is exact when
the population distribution is normal and is
approximately correct for large n in other cases.
Margin of Error
The margin of error for the population mean
when we use the data to estimate σ is
S
t*
n
The One-Sample t Test
Suppose that an SRS of size n is drawn
from a population having unknown mean µ.
To test the hypothesis Ho: µ = µ o based on
an SRS of size n, compute the one-sample t
statistic
x−µ
t=
s
n