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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