Power of a testPower of a test

The powerpower of a test (against a specific alternative value)• Is the probability that the test

will reject the null hypothesis when the alternative is true

• In practice, we carry out the test in hope of showing that the null hypothesis is false, so high power is important

H0 True

H0

FalseReject

Fail to reject

Type I Correct

Correct

Type II

Power

Suppose H0 is true – what if we decide to

fail to reject it?

Suppose H0 is false – what if we decide to

reject it?

Suppose H0 is true – what if we decide to reject

it?

Suppose H0 is false – what if

we decide to fail to reject it?

We correctly We correctly reject a false Hreject a false H00!!

A researcher selects a random sample of size 49 from a population with standard deviation = 35 in order to test at the 1% significance level the hypothesis:

H0: = 680

Ha: > 680

What is the probability of committing a Type I error? = .01

H0: = 680

Ha: > 680

For what values of the sample mean would you reject the null hypothesis?

Invnorm(.99,680,5) =691.63

H0: = 680

Ha: > 680

If H0 is rejected, suppose that a is 700. What is the probability of committing a Type II error?

What is the power of the test?

Normalcdf(-10^99,691.63,700,5) =.0471

Power = 1 - .0471 = .9529

H0: = 680

Ha: > 680

If H0 is rejected, suppose that a is 695. What is the probability of committing a Type II error?

What is the power of the test?

Normalcdf(-10^99,691.63,695,5) =.2502

Power = 1 - .2502 = .7498

Reject H0Fail to Reject H0

Power = 1 -

0

a

What happens to What happens to , , , & , & power when the sample size power when the sample size

is increased?is increased?Reject H0Fail to Reject H0

0

a

Facts:Facts:• The researcher is free to determine the

value of .• The experimenter cannot control , since it

is dependent on the alternate value.• The ideal situation is to have as small as

possible and power close to 1. (Power > .8)• As increases, powerpower increases. (But also

the chance of a type I error has increased!)• Best way to increase power, without

increasing , is to increase the sample sizesample size

A water quality control board reports that water is unsafe for drinking if the mean nitrate concentration exceeds 30 ppm. Water specimens are taken from a well.

Identify the decision:

a) You decide that the water is not safe to drink when in fact it is safe.

Type I Error

A water quality control board reports that water is unsafe for drinking if the mean nitrate concentration exceeds 30 ppm. Water specimens are taken from a well.

Identify the decision:

b) You decide that the water is not safe to drink when in fact it is not safe.

Correct –

Power!!

Bottles of a popular cola are suppose to contain 300 ml of cola. A consumer group believes the company is under-filling the bottles. (Assume = 50 with n = 30)

Find the power of this test against the alternative = 296 ml.