Hypothesis testing and the power of the test

[adeel]

I am having trouble understanding the concept of the power of the test. Here is a sample question with solution:

A company wants to test if average weekly demand is more than 2000 lbs. Test is to be carried out at 5% level of significance, and an estimate of the population variance is 1,000,000. What is the power of the test if the true mean is 2300 lbs.

So here is the sol'n (u represents population mean, and x represents x bar, sample mean, z is z-score, based on 5% is 1.645):

Hypothesis statement: Null: u <= 2000 Alternative: u > 2000

xcritical = u + zo (o is standard deviation, calculation shows it to be 200)
xcritical = 2000 + 1.645(200)
xcritical = 2349

Power at 2300

P(xcritical > 2349) = P (z > 0.145)
P(z > 0.145) = 0.5 - 0.0596 = 0.4404

So the thing i don't understand, is that if the power of the test is the probability of correctly rejecting the null hypothesis when it is false, why do we calculate the area beyond the z-score and call that the power of the test. Isnt the area beyond supposed to be Beta, the probability of making a type II error?


Any help is greatly appreciated

 

[EnumaElish]

http://linkage.rockefeller.edu/wli/glossary/stat.html#p
POWER
This is the probability that a statistical test will detect a defined pattern in data and declare the extent of the pattern as showing STATISTICAL SIGNIFICANCE. POWER is related to TYPE-2 ERROR by the simple formula : POWER = (1-BETA) ; the motive for this re-definition is so that an increase in value for POWER shall represent improvement of performance of a STATISTICAL TEST. For more detail, see : BETA.
BETA
Also known as TYPE-2 ERROR, BETA is the complement to POWER : BETA = (1-POWER). This is the probability that a statistical test will generate a false-negative error : failing to assert a defined pattern of deviation from a null pattern in circumstances where the defined pattern exists.

 

[tacman]

.

The power of a test is the probability of correctly rejecting the null hypothesis when it is false. In other words, it is the probability of correctly detecting a true effect or difference. In the example given, the true mean is 2300 lbs, which is greater than the hypothesized mean of 2000 lbs. This means that if the true mean is 2300 lbs, the null hypothesis (u <= 2000) is false.

The area beyond the z-score (0.145 in this case) is the probability of observing a sample mean greater than the critical value (2349) when the true mean is 2300 lbs. This is the same as the power of the test, because it represents the probability of correctly rejecting the null hypothesis (u <= 2000) when it is false (true mean is 2300).

The probability of making a type II error (incorrectly accepting the null hypothesis when it is false) is represented by the area between the critical value and the true mean. In this case, that area is 0.0596, which is the probability of making a type II error.

So, to summarize, the power of the test is the probability of correctly detecting a true effect or difference (represented by the area beyond the critical value), while the probability of making a type II error is represented by the area between the critical value and the true mean. Hope this helps clarify the concept of power in hypothesis testing.