Goals…Define what is meant by a Type I error. Define what is meant by a Type II error. Define what is meant by the power of a

test. Identify the relationship between the

power of a test and a Type II error. List four ways to increase the power of a

test.

Errors in Statistical Inferene

Type I ErrorWe make a “Type I Error” when we

incorrectly reject H0.

We make a “Type II Error” when we incorrectly fail to reject Ho.

TYPE I ERRORInform the EPA that the

water is safe to drink when it is actually UNSAFE!

TYPE II ERRORInform the EPA that the water is not safe to drink even though it really WAS

safe.

Consequence?

Consequence?

People drink unsafe water and could get

sick

People are unable to use this

valuable water source and the

government imposes water restrictions

• Which error do you believe is more serious? Why?

• If you had to choose and alpha level of α = 0.1, 0.05, or 0.01 which would you choose? Why?

TYPE I ERROR

Inform the EPA that the water is safe to drink when it is actually UNSAFE!

TYPE II ERRORInform the EPA that the water is not safe to drink even though it

really WAS safe.

People drink unsafe water and could get

sick

People are unable to use this valuable water source

and the government imposes water restrictions

Mel•Mel N. Colly is interested in whether or not his new treatment for depressed patients is decreasing his patients’ rating of depression. Suppose all of his depressed patients have a mean depression score of 8 with a standard deviation of 4. Mel chooses a random sample of 30 depressed patients treated with his innovative approach and determines that the mean depression score for these individuals is 7.5. Does the treatment decrease depression?

•H0: μ = 8•Ha: μ < 8

Mel •H0: μ = 8•Ha: μ < 8

•Describe and give the consequences of a Type I error and a Type II error.

•TYPE I ERROR• Description: Mel concludes that the mean depression score post-treatment is less than 8 even through it this is not the case. • Consequence: Mel uses the treatment on depressed patients even though it has no effect on them (or it could increase depression!) rather than researching a better treatment that could really help. Patients waste time and money on a useless (or harmful) treatment.

Mel •H0: μ = 8•Ha: μ < 8

•Describe and give the consequences of a Type I error and a Type II error.

•TYPE II ERROR• Description: Mel concludes that the mean

depression score post-treatment is still 8 (or higher), but in reality post-treatment scores are less than 8.

• Consequence: Mel does not use the effective treatment on his patients and these individual miss out on the opportunity to decrease their level of depression and improve their lives.

Mel

•Which error do you believe is more serious? Why?

•If you had to choose and alpha level of α = 0.1, 0.05, or 0.01 which would you choose? Why?

•TYPE I ERROR• Description: Mel concludes that the mean depression score post-treatment is less than 8 even through it this is not the case. • Consequence: Mel uses the treatment on depressed patients even though it has no effect on them rather than researching a better treatment that could really help. Patients waste time and money on a useless (or harmful) treatment.

•TYPE II ERROR• Description: Mel concludes that the mean depression score post-treatment is still 8 (or higher), but in reality post- treatment scores are less than 8.• Consequence: Mel does not use the

effective treatment on his patients and these individual miss out on the opportunity to decrease their level of depression and improve their lives.

Error Probabilities

H0 is true Ha is true

Reject H0

α (0.05)TYPE I ERROR

Correct Decision

Fail to reject H0Correct Decision

TYPE II ERROR

• We decide to use α = 0.05• What does that mean?

• The probability of a Type I error = α

Error ProbabilitiesH0 is true Ha is true

Reject H0

α

TYPE I ERROR

Correct Decision

Fail to reject H0Correct Decision

βTYPE II ERROR

• The probability of a Type II error is β

Error Probabilities

H0 is true Ha is true

Reject H0

α TYPE I ERROR

1-βCorrect Decision

Fail to reject H0Correct Decision

(1-α)

βTYPE II ERROR

• Which box do we WANT to fall in?

Error Probabilities

H0 is true Ha is true

Reject H0

α TYPE I ERROR

1-βCorrect Decision

Fail to reject H0Correct Decision

(1-α)

βTYPE II ERROR

• 1-β is called the POWER of the test

Error Probabilities

μ = 8(H0 is true)

μ < 8(Ha is true)

Reject H0

α =0.05TYPE I ERROR

Power = 1-β = 0.166

Correct Decision

Fail to reject H0Correct Decision

(1-α)

β = 0.834TYPE II ERROR

• Mel wants to use an alpha level of 0.05. • Find the probability of a Type I error• The probability of a Type II error is 0.834. What

is the Power of the test?

Power: You do NOT need to calculate β or Power by

hand (except in the way we just did in the chart). You just need to understand the concept.

Power is the probability that you will CORRECTLY reject the null hypothesis.

A perfect study would have VERY high powerThere are a few ways to increase power…http://bcs.whfreeman.com/tps3e/

HOMEWORK!!!

Classwork: Error and Power WorksheetHomework: 11.59, 11.62, 11.64, 11.69 (full test)