Student

1
Students t Distribution

• Lecture 34
• Section 10.2
• Mon, Apr 2, 2007

2
What if ? is Unknown?

• It is more realistic to assume that the value of
  ? is unknown.
• In this case, we use s to estimate ?.
• However, this changes everything.

3
Example

• See Example 10.1 on page 616 and assume that ? is
  unknown.
• Step 1 State the hypotheses.
• H0 ? 15 mg.
• H1 ? lt 15 mg.
• Step 2 State the significance level.
• ? 0.05.
• Step 3 What is the test statistic?
• We must digress.

4
What if ? is Unknown?

• Let us assume that the population is normal or
  nearly normal.
• Then the distribution of?x is normal.
• That is, for all sample sizes n,

5
What if ? is Unknown?

• Furthermore, for large n,
• However, for small n,

6
What if ? is Unknown?

• Why not?
• And if it is not N(0, 1), then what is it?

7
Students t Distribution

• It has a distribution called Students t
  distribution.
• The t distribution was discovered by W. S. Gosset
  in 1908.
• See http//mathworld.wolfram.com/Studentst-Distrib
  ution.html

8
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 ?.

9
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.