Today:
- Return tests
- Quiz (basics of the t-distribution)
- Section 8.2: Estimating
when
is unknown
- The t-distribution
- similar in shape to the normal
- we estimate
using the sample standard deviation, s:
whereas
- Properties of the t-distribution above:
- symmetric, like the normal
- centered at zero, like the normal
- had n-1 degrees of freedom
- the standard deviation of the distribution is
- Since the standard deviation of the distribution is
greater than 1, it is a little broader than the normal,
a little more variable.
- The major difference between the z- and t-distributions is that
for the normal we have one single table; for the t-distribution, we need to
know the value of n, and then use the table for that value of
df=n-1. (Rats!) Of course, with technology, this is not really much of a
problem.
- How does this affect our strategy? It's basically the same: see
page 290....
- Let's try an example:
- Section 9.1: Inferences comparing two populations
- Our over-riding objective is to decide whether two populations
share a parameter or not: e.g. is
the same for both?
- There are two kinds of samples of the populations we'll compare:
- Dependent (paired) samples
- Independent (unpaired) samples
- Dependent (paired) samples:
- husband versus wife hours in the kitchen
- male and female graduates matched on major and GPA
- "pre" and "post" intervention scores
- Independent (unpaired) samples:
- 100 students randomly chosen from sta205 at NKU
versus 100 students randomly chosen from the
equivalent of sta205 at EKU
- 1000 voters randomly chosen in KY versus 900 voters
randomly chosen in OH.
- One thing that separates the two pretty clearly is when the sample
sizes are different: it's almost a sure sign that the samples
are independent.
- Examples:
- Section 9.2: Inferences about two population means: independent samples
- Notation:
| Population 1 |
Mean:
|
Variance:
|
|
| Sample 1 |
Mean:
|
Variance:
|
Sample size:
|
| Population 2 |
Mean:
|
Variance:
|
|
| Sample 2 |
Mean:
|
Variance:
|
Sample size:
|
- We're going to be making inferences about
- Conditions under which we may do so (p. 330):
- Two independent random samples must be selected
- both sample means must be normally distributed
- Standard error of
:
- Confidence interval: usual story:
point estimate plus or minus t times the standard error
to
where the degrees of freedom are really horrible (p. 331) -- use
technology!
- Test of hypothesis: usual story:
Decision rule: Accept Ha if p-value <
Test Statistic:
with appropriate degrees of freedom (p. 331 -- use technology!)
- Examples (using StatCrunch):
- For the following exercises, do not "pool" the variances
(that is done when there is reasonable suspicion that the variances of the two
samples is the same -- we have no such suspicions)
- Example 9.3, p. 332 (http://www.nku.edu/~statistics/data/exam09-03.xls)
- #6, p. 341 (http://www.nku.edu/~statistics/data/c09s02e06.xls)
Links
Website maintained by Andy Long.
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