1. Summary of t-Test for Testing a Single Population Mean (m)

2. t-Test Statistic Assumptions:
Population is normal although this assumption can be relaxed if sample size is “large”.
Random sample was drawn from the population of interest.

3. The t-distribution Student(df ) density curves for various df.= [ i . e , N o r m a l ( 1 ) ] 5 2 4 -
1. The t-distribution has one parameter that controls it’s shape called the degrees of freedom.

4. The t-distribution d f = [ i . e , N o r m a l ( 1 ) ] 5 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).

5. The t-distribution d f = [ i . e , N o r m a l ( 1 ) ] 5 2 4 -
3. As df increases, the t-distribution becomes more and more like the standard normal.

6. The t-distribution d f = [ i . e , N o r m a l ( 1 ) ] 5 2 4 -
4. t-dist (df = ¥) and Normal (0, 1) are two ways of describing the same distribution.

7. estimate ± (t-quantile value) SE(estimate)
The t-distribution
From now on we will treat as having a t-distribution (df = n - 1).
For confidence intervals we will build t-standard-error intervals,
estimate ± (t-quantile value) SE(estimate)

8. The t-distribution Example:
P(-1.96 £ Z £ 1.96) = (standard normal)
P( £ 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

9. Form of Hypotheses Ho: m = mo HA: m < mo (lower-tail test)
HA: m > mo (upper-tail test)
t is negative t
P-value t
P-value
t is positive
P-values are computed by finding areas beneath a t-distribution (df = n – 1)

10. Form of Hypotheses
Ho: m = mo
HA: m mo (two-tailed test) t -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.

11. t-Probability Calculator in JMP
Enter test statistic value ( t ) and df in these cells and the tail probabilities will update automatically.

12. 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.