AMS312 Spring_2008

Apr 11th

Chapters 6&7 (continued):

Approach 1 – Pivotal Quantity Method

1、Inference on one population proportion p for large sample test: summary:

Same:

Same test statistic when Ho is true

At the significance level, we reject Ho in favor of Ha if:

Definition: P-value (observed significance level) is the probability to observe the test statistic value or values more extreme than the test statistic value (here being Zo), given that Ho is true.

* Make conclusion using the P-value.

At the significance level, we reject Ho in favor of Ha if the P-value is less than.

* P-value is more informative.

2、Inference on one population meanfor large sample test: Summary:

Same:

Same Test statistic:

Pivotal quantity: when is unknown.

Whenis known.

(*Z follows exactly N(0,1) distribution when the population is normal)

Test statistic: or

At the significance level, we reject Ho if:

P-values are the same as above (proportion)

Derivation for the 2-sided test: P (Type 1 error)=P (reject Ho|Ho)

Hence

Therefore, we reject Ho atif:

Example 1 (Problem 6.2.10 from Page439)

At a class research project, Rosaura wants to see whether the stress of final exams elevates the blood pressures of freshmen women. When they are not under any untoward duress, healthy 18-year-old women have systolic blood pressures that average 120mmHg. If Rosaura finds that the average blood pressure for the 50 women in statistics 101 on the day of the final exam is 125.2 with sample standard deviation 13 mm Hg, what should she conclude? Set up and test an appropriate hypothesis.

Solution: This is test on one population mean for large sample.

N=50,,

Since Zo=2.8>1.645, we reject Ho at

P-value=0.0026, since P-value<0.05, we reject Ho

If: since now P-value is still smaller than 0.01, we reject Ho.

3、Inference on one population mean, small sample test, normal population:

Same:

Now the pivotal quantity is:

Exactly

William S. Gosset (1876 -1937), inventor of the Student’s t-distribution.

Example 2 (Page 490): Pica is a children’s disorder characterized by a craving for nonfood substances such as clay, plaster, and paint. Anyone affected runs the risk of ingesting high levels of lead, which can result in kidney damage and neurological dysfunction. Checking a child’s blood lead level is a standard procedure for diagnosing the condition.

Among children between the ages of six months and five years, blood lead levels of 16.0mg/l are considered “normal”. Recently, a random sample of twelve children enrolled in Molly’s Mighty Bear Nursery had their blood lead levels checked. The resulting sample mean and samplestandard derivation were 18.65 and 5.049, respectively. Can it be concluded that children at this particular facility tend to have higher lead levels? At thelevel, is the increase from 16.0 to 18.65 statistically significant?

Solution: Average lead level for all children in Milly’s daycare

, s=5.049 ()

Assuming the blood lead distribution for the population is normal

Test statistic:

At, we will reject the null hypothesis if

Now 1.82>1.7959 hence we reject the null hypothesis.

Approach 2 -- Likelihood ratio test (LRT):

Data:: i.i.d. follow p.d.f. f (x;)

Let

Likelihood function:

If Ho is true, then so we should reject Ho if

Use the definition of Type I error rate (significance level) to derive the threshold value c as usual. That is: Set and then solve for c

Example 3: Let ,is unknown. Please derive the LRT for.

MLE’s forandin

For the numerator:

(*)

Then

Hence

Similarly, we can derive:

For the denominator:

Therefore,

At the significance level , we reject the null hypothesis if

c is the value such that:

And,

Hence the above formula becomes:

Given

Symmetric, therefore we reject Ho at significance level if:

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