Summary of t-Test for Testing a
Single Population Mean (m)
t-Test Statistic
Assumptions:
1. Population is normal although this assumption can be
relaxed if sample size is “large”.
2. Random sample was drawn from the population of
interest.
The t-distribution
Student(df ) density curves for various df.
df = 
[i.e., Normal(0,1)]
df = 5
df = 2
- 4
- 2
0
2
4
1. The t-distribution has one parameter that
controls it’s shape called the degrees of freedom.
The t-distribution
df = 
[i.e., Normal(0,1)]
df = 5
df = 2
- 4
- 2
0
2
4
2. The Student’s t-distribution is bell shaped and
centred at 0 — like the Standard Normal
distribution but more variable (larger spread).
The t-distribution
df = 
[i.e., Normal(0,1)]
df = 5
df = 2
- 4
- 2
0
2
4
3. As df increases, the t-distribution becomes
more and more like the standard normal.
The t-distribution
df = 
[i.e., Normal(0,1)]
df = 5
df = 2
- 4
- 2
0
2
4
4. t-dist (df = ) and Normal (0, 1) are two ways
of describing the same distribution.
The t-distribution
X  mo
From now on we will treat
as
SE X 
having a t-distribution (df = n - 1).
For confidence intervals we will build
t-standard-error intervals,
estimate ± (t-quantile value) SE(estimate)
The t-distribution
Example:
P(-1.96 Z ) = 0.95 (standard normal)
P(-2.365 t 2.365) = 0.95 for t-dist. w/ df = 7
Hence, if we are taking samples of size n = 8 and we
want to build intervals that include m for 95% of all
samples taken in the long run, then we use
Form of Hypotheses
Ho: m=mo
HA:m<mo(lower-tail test)
P-value
t
0
t is negative
Ho: m=mo
HA:m>mo(upper-tail test)
P-value
0 t
t is positive
P-values are computed by finding areas
beneath a t-distribution (df = n – 1)
Form of Hypotheses
Ho: m=mo
HA:mmo(two-tailed test)
-t
0 t
P-value = Shaded Area
t is either pos. or neg.
This test is equivalent to constructing a
100(1-a)% CI for m and checking in mo is
contained in the resulting interval.
Reject Ho if the CI does not cover mo.
t-Probability Calculator in JMP
Enter test statistic value ( t ) and df in these
cells and the tail probabilities will update
automatically.
t-Quantile Calculator for CI’s
The t-table value or standard
error multiplier for the desired
confidence level appears here.
Enter desired confidence level
which is typically 90, 95, or 99.