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Power of a test

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Power of a test

- 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

Suppose H0 is false – what if we decide to reject it?

Suppose H0 is false – what if we decide to fail to reject it?

We correctly reject a false H0!

Suppose H0 is true – what if we decide to fail to reject it?

Type I

Correct

a

Power

Suppose H0 is true – what if we decide to reject it?

Correct

Type II

b

A new flu vaccine claims to prevent a certain type of flu in 70% of the people who are vaccinated. A random sample of 100 people are vaccinated and then exposed to the flu. Is this claim too high? Use a = .05.

What are the hypotheses?

H0: p = .7

Ha: p < .7

A new flu vaccine claims to prevent a certain type of flu in 70% of the people who are vaccinated. A random sample of 100 people are vaccinated and then exposed to the flu. Is this claim too high? Use a = .05.

Find mp and sp.

mp = .7

sp = .0458

A new flu vaccine claims to prevent a certain type of flu in 70% of the people who are vaccinated. A random sample of 100 people are vaccinated and then exposed to the flu.Is this claim too high? Use a = .05.

What is the probability of committing a Type I error?

a = .05

.7

H0: p = .7

Ha: p < .7a = .05

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

So if we get p-hat=.625 or less, we would reject H0.

a = .05

p?

Invnorm(.05,.7,.0458) =.625

.7

.6

Where did this number come from?

H0: p = .7

Ha: p < .7

We reject H0 and decide that p<.7.

Suppose that pa is 0.6.

What is the probability of committing a Type II error?

I selected a number that was less than .7

What is a type II error?

How can we find this area?

a = .05

Reject

failing to reject H0 when the alternative is true

What is the standard deviation of this curve?

b = ?

Normalcdf(.625,∞,.6,.0490) =.305

a = .05

.7

.6

b =.305

What is the power of the test?

What is the definition of power?

Power - the probability that the test correctly rejects H0, if p = .6, is .695

Is power a conditional probability?

The probability that the test correctly rejects H0

Power = ?

Power = 1 - .305= .695

a = .05

.7

.6

.55

- Suppose we select .55 as the alternative proportion (p).
- What is the probability of the type II error?
- b) What is the power of the test?

What happened to the power of the test when the difference |p0 – pa| is increased?

b = normalcdf(.625,∞, .55,.0497) = .066

Power = 1 - .066= .934

a = .05

.65

.7

.6

Suppose we select .65 as the alternative proportion (p).

a) What is the probability of the type II error?

b) What is the power of the test?

What happened to the power when the difference |p0-pa| is decreased?

b = normalcdf(.625,∞, .65,.0477) = .700

b

Power

Power = 1 - .700= .300

.7

.6

Suppose that we change alpha to 10%.

Using pa = .6, what would happen to the probability of a type II error and the power of the test?

a = .1

a = .05

b = .2000

Power = .8000

The probability of the type II error (b) decrease and power increased, BUT the probability of a type I error also increased.

Power

b

Fail to Reject H0

Reject H0

Power increases when n increases

a

p0

P(type II) decreases when n increases

Power

b

pa

p0

pa

Fail to Reject H0

Reject H0

a

Power = 1 -b

b

What affects the power of a test?

As |p0 – pa| increases, power increases

As a increases, power increases

As n increases, power increases

- The researcher is free to determine the value of a.
- The experimenter cannot control b, since it is dependent on the alternate value.
- The ideal situation is to have a as small as possible and power close to 1. (Power > .8)
- Asa increases, power increases. (But also the chance of a type I error has increased!)
- Best way to increase power, without increasing a, is to increase the sample size

A new flu vaccine claims to prevent a certain type of flu in 70% of the people who are vaccinated. In a test, vaccinated people were exposed to the flu. Is this claim too high?

Identify the decision:

a) You decide that the proportion of vaccinated people who do not get the flu is less than 70% when it really is not.

Type I Error

A new flu vaccine claims to prevent a certain type of flu in 70% of the people who are vaccinated. In a test, vaccinated people were exposed to the flu. Is this claim too high?

Identify the decision:

b) You decide that the proportion of vaccinated people who do not get the flu is less than 70% when it really is.

Correct – Power!!