STAT 200
Chapter 7
Inference for Distributions
Inferences based on the tdistribution
(Section 7.1)
•
Let
X
be a random variable that follows a certain distribution with mean
μ
and
standard deviation
σ
. If we have a large enough sample size
n
, and other necessary
conditions being satisfied, then the sampling distribution of the sample means
X
follows
approximately
N
(
μ,
σ
√
n
). If the underlying distribution of
X
is normal, the sampling
distribution is normal regardless of the sample size.
•
In most situations, the value of the population standard deviation
σ
is unknown. We
will estimate
σ
using the sample standard deviation
s
from a random sample, and
estimate
SD
(
X
) =
σ
√
n
by the standard error
SE
(
X
) =
s
√
n
.
•
For any particular
X
value, the corresponding zscore
Z
=
X

μ
SD
(
X
)
follows approxi
mately the standard normal distribution.
When
σ
is unknown and
SE
(
X
) is used
to estimate
SD
(
X
), the quantity
X

μ
SE
(
X
)
is no longer well modelled by the standard
normal distribution.
The sampling distribution of
X

μ
SE
(
X
)
(we obtain one
x
for each
repeated sample of a fixed sample size
n
) has thicker tails than the standard normal
distribution. Also, the shape of the distribution changes with the sample size. We use
“
T
” to denote this quantity,
T
=
X

μ
SE
(
X
)
. The sampling distribution of
T
is called the
Student’s
t
distribution
.
•
Properties of the
t
distribution
–
perfectly symmetric about the mean 0
–
unimodal, bellshaped
–
has one parameter – the degrees of freedom (
df
) which determines the shape of
the distribution and is given by
n

1
–
has thicker tails when sample size is smaller
–
approaches the standard normal distribution as sample size increases
Confidence intervals for the population mean
μ
•
When
σ
is unknown, we’ll use
SE
(
X
) and the
t
distribution to construct confidence
intervals for
μ
.
A confidence interval of confidence level
C
for
μ
is constructed using:
1
x
±
t
*
n

1
s
√
n
where the critical point
t
*
n

1
is the “
t
score” such that the area to its right under the
t
curve with
n

1 degrees of freedom is equal to
100%

confidence level
2
(See Table D in your textbook).
Such a confidence interval for
μ
is called the onesample
t
confidence interval
.
•
Assumptions and conditions for using the tdistribution for statistical inferences
–
the sample is randomly drawn from the population
–
the sampled values are independent (sample size is a small fraction of the popu
lation size)
–
when the underlying distribution of
X
is nearly normal, or unimodal and sym
metric, using the
t
distribution is justified even if sample size is small. When the
underlying distribution is skewed or nonnormal, we will need a large sample for
the
t
distribution to work well.
Hypothesis testing for the population mean
μ
– the onesample
t
test
1. Different forms of a hypothesis test on a population mean
μ
:
H
0
:
μ
=
μ
o
for some fixed
μ
o
vs. (1)
H
a
:
μ
=
μ
o
(twosided test)
(2)
H
a
:
μ > μ
o
(onesided, righttailed test)
(3)
H
a
:
μ < μ
o
(onesided, lefttailed test)