1. Student’s t Distribution
Lecture 33
Section 10.2
Wed, Dec 1, 2004

2. What if  is Unknown?
It is more realistic to assume that the value of  is unknown.
(If we don’t know the value of , then we probably don’t know the value of ).
In this case, we use s to estimate .

3. What if  is Unknown?
Let us assume that the population is normal or nearly normal.
Then the distribution ofx is normal.
That is,x is N(, /n).
However,x is not N(, s/n) unless the sample size is large enough, (n  30).

4. What if  is Unknown? In other words,
is not standard normal, so we can’t use the tables.
If it is not N(0, 1) , then what is it?

5. Student’s t Distribution
It has a distribution called Student’s t distribution.
The t distribution was discovered by W. S. Gosset in 1908.
He used the pseudonym “Student” to avoid getting fired for doing statistics on the job!!!

6. The t Distribution
The shape of the t distribution is very similar to the shape of the standard normal distribution.
However, the t distribution has a (slightly) different shape for each possible sample size.
They are all symmetric and unimodal.
They are all centered at 0.

7. The t Distribution
They are somewhat broader than Z, reflecting the additional uncertainty resulting from using s in place of .
As n gets larger and larger, the shape of the t distribution approaches the standard normal.

8. Degrees of Freedom
If the sample size is n, then t is said to have n – 1 degrees of freedom.
We use df to denote “degrees of freedom.”

9. Standard Normal vs. t Distribution
The distributions t(2), t(30), and N(0, 1).
t(2)
t(30)
N(0, 1)

10. Table IV – t Percentiles
Table IV gives certain percentiles of t for certain degrees of freedom.
Specific percentiles for upper-tail areas:
0.40, 0.30, 0.20, 0.10, 0.05, 0.025, 0.01,
Specific degrees of freedom:
1, 2, 3, …, 30, 40, 60, 120.

11. Table IV – t Percentiles
The table tells us, for example, that
P(t > 1.812) = 0.05, when df = 10.
Since the t distribution is symmetric, we can also use the table for lower tails by making the t values negative.
So, what is P(t < –1.812), when df = 10?

12. Table IV – t Percentiles
The table allows us to look up certain percentiles, but it will not allow us to find probabilities.

13. TI-83 – Student’s t Distribution
The TI-83 will find probabilities for the t distribution
Press DISTR.
Select tcdf and press ENTER.
tcdf( appears in the display.
Enter the lower endpoint.
Enter the upper endpoint.

14. TI-83 – Student’s t Distribution
Enter the number of degrees of freedom (n – 1).
Press ENTER.
The result is the probability.

15. Example Enter tcdf(1.812, 99, 10). The result is 0.0500.
Thus, P(t > 1.812) = 0.05 when there are 10 degrees of freedom (n = 11).

16. TI-83 – Student’s t Distribution
The TI-83 allows us to find probabilities, but it will not find percentiles for the t distribution.

17. Hypothesis Testing with t
We should use the t distribution if
The population is normal (or nearly normal), and
 is unknown, so we use s in its place, and
The sample size is small (n < 30).
Otherwise, we should not use t.

18. Hypothesis Testing with t
The hypothesis testing procedure is the same except for two steps.
Step 3: Find the value of the test statistic.
The test statistic is now
Step 4: Find the p-value.
We must look it up in the t table, or use tcdf on the TI-83.

19. Example
Re-do Example 10.1 (by hand) under the assumption that  is unknown.

20. TI-83 – Hypothesis Testing When  is Unknown
Press STAT.
Select TESTS.
Select T-Test.
A window appears requesting information.
Choose Data or Stats.

21. TI-83 – Hypothesis Testing When  is Unknown
Assuming we selected Stats,
Enter 0.
Enterx.
Enter s. (Remember,  is unknown.)
Enter n.
Select the alternative hypothesis and press ENTER.
Select Calculate and press ENTER.

22. TI-83 – Hypothesis Testing When  is Unknown
A window appears with the following information.
The title “T-Test”
The alternative hypothesis.
The value of the test statistic t.
The p-value.
The sample mean.
The sample standard deviation.
The sample size.

23. Example
Re-do Example 10.1 on the TI-83 under the assumption that  is unknown.

24. Let’s Do It! Let’s do it! 10.3, p. 582 – Study Time.
Let’s do it! 10.4, p. 583 – pH Levels.