AMS 572 Lecture Notes
AMS 572 Lecture Notes
Sept. 27, 2007.
One-sample Inference on Population Mean
④ When the population in normal, and the population variance known
Data : [pic]
Hypothesis test [pic]
- Pivotal Quantity Method (one-sided or one-tailed test)
“Standard Procedures”
* pivotal quantity for [pic] [pic]
* Test statistic (plug in [pic] to Z)
[pic] ([pic], [pic])
* Derive the decision threshold for your test based on the Type I error rate the significance level [pic]
* [pic] = P(Type I error) = P(reject [pic]
= P([pic]
[pic]
[pic]
* We reject [pic] in front of [pic] if [pic]
Other Hypotheses
[pic] (one-sided test or one-tailed test)
Test statistic : [pic]
[pic]
[pic]
[pic] (Two-sided or Two-tailed test)
Test statistic : [pic]
[pic]
[pic]
[pic]
[pic]
Reject [pic] if [pic]
- We have just discussed the “rejection region” approach for decision making.
- There is another approach for decision making, it is “p-value” approach.
* p-value : the probability that we observe a test statistic value that is as extreme, or more extreme, than the one we observed.
|[pic] |[pic] |[pic] |
|Observed value of test statistic [pic] |
| | |p-value[pic] |
|p-value[pic] |p-value[pic] |[pic] |
|(1) the area under N(0,1) pdf to the right |(2) the area under N(0,1) pdf to the left |(3) twice the area to the right of [pic] |
|of [pic] |of [pic] | |
(1)
[pic]
(2)
[pic]
(3)
[pic]
The way we make conclusions in the same for all hypotheses.
“We reject [pic] in front of [pic] iff p-value[pic]”
⑤ Any population, the population variance is unknown
However – the sample size is large [pic]
- the “large sample” scenario
Thm The Central Limit Theorem
Let [pic] be a random sample from a population with mean [pic] and variance [pic]
[pic]
* When r is large enough [pic],
[pic] (approximately)
P.Q. for ⑤
|[pic] |[pic] |[pic] |
|Test Statistic [pic] |
|Rejection region : we reject [pic] in favor of [pic] at the significance level [pic] if |
|[pic] |[pic] |[pic] |
|(1) |(2) |(3) |
(1)
[pic]
(2)
[pic]
(3)
[pic]
p-value = 2 * this area
⑥ Normal Population, but the variance is unknown
100 years ago – people use Z-test
This is OK for n large [pic] ⑤ CLT
This is NOT ok if the sample size is samll.
“A Student of Statistics” – pen name of William Gossett (June 13, 1876)
“The Student’s t-test”
P.Q. [pic]
( Exactly t-distribution with n-1 degrees of freedom )
Wrong Test
Reject [pic]
[pic]
Right Test
Reject [pic]
[pic]
[pic]
(Because t distribution has the heavier tail than normal distribution.)
Right Test
* Test Statistic [pic]
[pic]
* Reject region : Reject [pic] at [pic] if [pic]
* p-value
[pic]
Area * 2 = p-value
- Definition : t-distribution
[pic]
[pic]
[pic] (chi-square distribution with k degree of freedom)
[pic] are independent.
- Def 1 : chi-square distribution : from the definition of the gamma distribution
(chi-square distributio is a special gamma distribution.)
- Def 2 : chi-square distributino : Let [pic],
then [pic]
Normal Distribution
Gaussian Distribution
Carl Friedrich Gauss (April 30, 1977, German)
Abraham de Moivre (May 26, 1667, French) : real founder
Thm Sampling from the normal population
Let [pic], then
① [pic]
② [pic]
③ [pic] and [pic] are independent.
[pic]
Proof) [pic]
[pic]
Let [pic]
[pic]
- Review of Order Statistics
Random sample : [pic]population with pdf f(x)
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
Example Jerry is planning to purchase a sports good store. He calculated that in order to cover basic expenses, the average daily sales must be at least $525.
Scenario A. He checked the daily sales of 36 randomly selected business days, and found the average daily sales to be $565 with a standard deviation of $150.
Scenario B. Now suppose he is only allowed to sample 9 days. And the 9 days sales are $510, 537, 548, 592, 503, 490, 601, 499, 640.
For A and B, please determine whether Jerry can conclude the daily sales to be at least $525 at the significance level of [pic]. What is the p-value for each scenario?
Solution A large sample (⑤) n=36, [pic]
[pic] versus [pic]
Test statistic [pic]
At the significance level [pic], we will reject [pic] if [pic]
[pic] We can not reject [pic]
p-value
[pic]
p-value = 0.0548
Alternatively, if you can show the population is normal using the Shapiro-Wilk test, you can perform a t-test.
Solution B small sample [pic] Shapiro-Wilk test
[pic] If the population is normal, t-test is usable.
[pic]
[pic] versus [pic]
Test statistic [pic]
At the significance level [pic], we will reject [pic] if [pic]
[pic] We can not reject [pic]
p-value
[pic]
What’s p-value when [pic]?
Topics in next lecture
① Power of the test
② Sample size determinants
③ Do it in SAS
④ Inference on [pic]
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