AMS 572 Lecture Notes 4

AMS 572 Lecture Notes 4

Sep. 23rd, 2008

P-value = P(observing a sample/test statistic as extreme or more extreme than the one observed |).

Normal population with known population variance.

That is, , in which, is known.

Two approaches to make decision for your test:

(1)Rejection region method(R.R method)

(2)P-value method

Test statistic:

(1)R.R method: At the significance level , reject if

(2)P-value method: At the significance leve , reject if P-value=

Test statistic:

(1)R.R method: At the significance level , reject if

(2)P-value method: At the significance leve , reject if P-value=

Test statistic:

(1)R.R method: At the significance level , reject if

(2)P-value method: At the significance level, reject if P-value=

For , prove that the two methods are equivalent.

Proof: (1) (2)

(1)At the significance level , reject if

P-value=

(2)(1)

(2) At the significance level ,reject if P-value=

P-value==

e.g About the average height of adult U.S male.

n=100, =65 (in), =2.5 (in)

T.S :

== 4

P-value=<0.0002

At =0.05, we reject , since P-value<0.05;

At =0.01, we reject , since P-value<0.01;

At =0.005, we reject , since P-value<0.005;

Power calculation.

e.g John Pauzke, president of Cereal’s Unlimited Inc, wants to be very certain that the mean weight of packages satisfies the package label weight of 16 ounces. The packages are filled by a machine that in set to fill each package to a specified weight. However, the machine has random variability measured by . John would like to have strong evidence that the mean package weight is about 16 oz. George Williams, quality control manager, advises him to examine a random sample of 25 packages of cereal. From his past experience, George knew that the weight of the packages follows a normal distribution with standard deviation 0.4 oz. At the significance level,

(1)What is the decision rule (rejection region) in terms of the sample mean?

(2)What is the power of the test when 16.2 oz?

(3)How many packages of cereal should be sampled if we wish to achieve a power of 85% when 16.2 oz?

Sep. 25th, 2008

Solution: Let be the weight of the i-th randomly selected package. Then,

(1)

Test Statistic : if

We reject at if (oz)

(2)(n=25)

Power = P(Reject )

(3)Power=1-

Figure1: Illustration of the derivation of power for one-sided test.

*Sample size determination based on the length of the C.I and based on the maximum error.

Question (4): What sample size do we need to be 95% sure that the discrepancy between the sample mean and the population mean is within 0.5 ounce?

(4)

General formula: