1. Student’s t Distribution
Lecture 35
Section 10.2
Tue, Apr 3, 2007

2. The t Distribution
The shape of the t distribution is very similar to the shape of the standard normal distribution.
It is
symmetric
unimodal
centered at 0.
But it is wider than the standard normal.
That is because of the additional variability introduced by using s instead of .

3. The t Distribution
Furthermore, the t distribution has a (slightly) different shape for each possible sample size.
As n gets larger and larger, s exhibits less and less variability, so the shape of the t distribution approaches the standard normal.
In fact, if n  30, then the t distribution is approximately standard normal.

4. 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.”
We will use the notation t(df) to represent the t distribution with df degrees of freedom.
For example, t(5) is the t distribution with 5 degrees of freedom (i.e., sample size 6).

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

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

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

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

9. The Decision Tree Is  known? yes no Is the population normal? TBA yes
Is n  30?
yes no
Give up

10. The Decision Tree Is  known? yes no Is the population normal?
Is n  30?
yes no
Give up

11. The Decision Tree Is  known? yes no Is the population normal?
Is n  30?
Is n  30?
yes no
yes no
Give up

12. The Decision Tree Is  known? yes no Is the population normal?
Is n  30?
Is n  30?
yes no
yes no
Give up

13. The Decision Tree Is  known? yes no Is the population normal?
Is n  30?
Is n  30?
yes no
yes no
Give up

14. The Decision Tree Is  known? yes no Is the population normal?
Is n  30?
Is n  30?
Is n  30?
yes no
yes no
yes no
Give up

15. The Decision Tree Is  known? yes no Is the population normal?
Is n  30?
Is n  30?
Is n  30?
yes no
yes no
yes no
Give up

16. The Decision Tree Is  known? yes no Is the population normal?
Is n  30?
Is n  30?
Is n  30?
yes no
yes no
yes no
Give up
Give up

17. When to Use Z-Test Is  known? yes no Is the population normal?
Is n  30?
Is n  30?
Is n  30?
yes no
yes no
yes no
Give up
Give up

18. When to Use T-Test Is  known? yes no Is the population normal?
Is n  30?
Is n  30?
Is n  30?
yes no
yes no
yes no
Give up
Give up

19. When to Use Either Z-Test or T-Test
Is  known?
yes no
Is the population
normal?
Is the population
normal?
yes no
yes no
Is n  30?
Is n  30?
Is n  30?
yes no
yes no
yes no
Give up
Give up

20. When to Give Up Is  known? yes no Is the population normal?
Is n  30?
Is n  30?
Is n  30?
yes no
yes no
yes no
Give up
Give up

21. 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.

22. Table IV – t Percentiles
What is the 95th percentile of the t distribution when df = 10?
The table tells us that
P(t > 1.812) = 0.05, when df = 10.
Thus, the 95th percentile is

23. Table IV – t Percentiles
Since the t distribution is symmetric, we can also use the table for lower tails by making the t values negative.
So, what is the 5th percentile of t when df = 10?
What is the 90th percentile of t when df = 25?

24. Table IV – t Percentiles
The table allows us to look up certain percentiles, but it will not allow us to find probabilities in general.
For example, what is P(t > 2.4) when df = 12?

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

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

27. Examples Enter tcdf(1.812, E99, 10). Enter tcdf(-E99, -1.812, 10).
The result is
Enter tcdf(-E99, , 10).
Thus, P(t > 1.812) = 0.05 when there are 10 degrees of freedom (n = 11).
Find P(t > 2.4) when df = 12.

28. 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.
Either use Z or “give up.”

29. Hypothesis Testing with t
Continue with Example 10.1, p. 616.
Step 3: Find the value of the test statistic.
The test statistic is now
Step 4: Compute t fromx, , s, and n.

30. Example Step 5: Find the p-value.
We must look it up in the t table, or use tcdf on the TI-83.
Step 6: Make the decision regarding H0 and H1.
Step 7: State the conclusion about cigarettes and carbon monoxide.

31. Example H0:  = 15 mg H1:  < 15 mg = 0.05.
p-value = P(t < ) = tcdf(-E99, , 24) =
Reject H0.
The carbon-monoxide content of cigarettes today is less than it was in the past.

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

33. 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.

34. 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.

35. Example
Re-do Example 10.1, p. 616, on the TI-83 under the assumption that  is unknown.

36. Practice
Use the TI-83 to find the following probabilities for the t distribution.
P(T > 2.5), df = 14.
P(-3 < T < 3), df = 7.
P(T < 3.5), df = 5.
Use the TI-83 to find the following t-distribution percentiles.
95th percentile, df = 25.
20th percentile, df = 21.
Endpoints of the middle 80%, df = 12.