1
NOTES: t-distributions
Hypothesis tests and confidence intervals involving mean
µ
with unknown
σ
.
Suppose that hypothesis tests and confidence intervals involving
µ
are provided with a known
standard deviation of the population
(
)
σ
. This would enable us to calculate the standard deviation of the
sample mean distribution since
( )
n
xSD
σ
=
, where
n
is the sample size. We now assume that
σ
is
not known (as is most generally the case in real life), and therefore replace
(
)
xSD
with the Standard Error
of the Mean. In other words, we use
( )
n
S
x
x
=SE
, where
x
S
is the standard deviation of the sample.
This, however, causes a problem since the
x
S
fluctuates with the sample size
n
in such a way as to create
a different (possible not normal) distributions for each sample size
n
. Each of these distributions is called
a t-distribution.
Facts about t-distributions:
• bell-shaped curve
• mean = 0
• symmetric about 0
• Each one is different from the standard normal curve. As
n
gets larger the
t-distributions get closer to the standard normal curve. When
30
≥
n
the
t-distributions are almost normal (are very close to the standard normal curve).
• degrees of freedom =
1
−
n
. The degrees of freedom (df) indicate which
t-distribution you will use.
For hypothesis tests, we will use a “t-test statistic” or “t-score” instead of a z-score. The equation for the t-
score is
n
S
x
t
x
0
µ
−
=
, where
0
µ
is the assumed population mean from the null hypothesis
0
H
. You will
also need to use tcdf instead of normalcdf on your calculator. 2
ND
VARS tcdf. Inputs are tcdf(LB, UB, df).
For confidence intervals we will use
*t
instead of
*
z
. The
*t
value depends on the t-distribution you
are using, and so depends on the degrees of freedom (
1
−
n
).
*t
cannot be obtained from your graphing
calculator and so it must be looked up on a table in your textbook. The formula for a confidence interval
for means (with unknown
σ
) is
n
S
tx
x
*
±
.
Requirements for using t-distributions:
1. SRS: The sample mean must be chosen from a random sample.
2. Sufficiently large sample size:
a. CASE:
15
<
n
. The data should be very close to a Normal model. Do not use t-
methods if there is strong skewness or outliers.
b. CASE:
4015
<
≤
n
. t-methods should work as long as the data is unimodal and
reasonably symmetric (make a histogram). t-methods should not be used in the presence
of outliers or strong skewness.
c. CASE:
n
≤
40
. t-methods can be used even in the presence of strong skewness or a
few outliers. In this case t-methods are called “Robust.”