1. Errors in Hypothesis Tests

2. Hypothesis test decisions:
reject H0 or fail to reject H0
Is there a possibility that our decision is wrong?
How many ways could we be wrong?

3. Type I Error
Reject a true H0
P(Type I Error) = α
Significance level! α μ0

4. Type II: fail to reject a false H0
Type II Error
Fail to reject a false H0
P(Type II Error) = β
Type I: reject a true H0
Fail to reject
Reject μa μ0
x-bar is really part of the H0 curve, but we decide it's part of the Ha curve α
x-bar is really part of the Ha curve, but we decide it's part of the H0 curve
Type II: fail to reject a false H0 β

5. H0 True H0 False Reject Fail to Reject Type I Error Type II Error
Correct
(α)
Type II Error
Correct
Suppose H0 is false & we reject it. What type of decision was made?
Suppose H0 is false & we fail to reject it. What type of decision was made?
Suppose H0 is true & we fail to reject it. What type of decision was made?
Suppose H0 is true & we reject it. What type of decision was made?
(β)

6. Explaining Type I & II Errors
“We decide this,
when really this is true.”
Just replace the red underlined words with context

7. Explaining Type I & II Errors
Consequence: A result of the error
 NOT the error itself

8. Consider a murder trial.
Our society believes Type I is worse – that's why we need sufficient evidence, beyond a reasonable doubt
H0: defendant is innocent
Ha: defendant is guilty
What are the hypotheses?
Type I Error:
Consequence:
Type II Error:
Decide the defendant is guilty when really innocent
An innocent person goes to prison
Decide defendant is not guilty when really guilty
A guilty person goes free
Which error has a more serious consequence?

9. Why not always use a tiny α?
As α increases, β decreases
As α decreases, β increases
Every decision is potentially an error
α & β are inversely related
Fail to reject H0
Reject H0 α μ0 β μa

10. Someone made an error with these products
Some Considerations
Seriousness of errors is determined by the context
Neither type is always more serious
We usually DO NOT know if an error is made in real life
Exceptions: Recalls
Firestone tires in 2000 (treads peeled off – 175 deaths, 700 injuries)
Vioxx in 2004 (arthritis drug – 27,000 lawsuits over heart attacks & strokes)
Someone made an error with these products

11. Sometimes the seriousness depends upon the person’s point of view
Lay’s Chip Company decides to accept a truckload of potatoes based on positive results from a sample of potatoes.
What are the hypotheses?
Type I error?
Type II error?
From the supplier’s viewpoint, which is more serious?
From the chip company’s viewpoint, which is more serious?
H0: potatoes good
Ha: potatoes bad
Decide the potatoes are bad when they really are good
Decide the potatoes are good when they really are bad
Sometimes the seriousness depends upon the person’s point of view
Type I
Type II

12. What are the hypotheses? Type I and II errors?
Water samples are taken from water used for cooling as it is being discharged from a power plant into a river. As long as the mean temperature of the discharged water is at most 150 degrees F, there will be no negative effects on the river’s ecosystem. To investigate whether the plant is in compliance with these regulations, fifty water samples will be taken at randomly selected times, and the temperature of each sample will be recorded.
What are the hypotheses?
Type I and II errors?
Which is more serious?
H0: μ = 150
Ha: μ >150
where μ is the true mean temperature
Type I: Decide temp. is above 150° when it’s really below
Type II: Decide temp. isn’t above 150° when it’s really above
Probably Type II – it would endanger the river's ecosystem

13. A doctor is considering a new medication to help fight infections
A doctor is considering a new medication to help fight infections. However, the medication has the possibility of being highly toxic to the patient. You will test the medication to determine toxicity.
Hypotheses?
Errors?
Which is more serious?
H0: medicine is not toxic
Ha: medicine is toxic
Type I: decide medicine is toxic when it really isn’t
Type II: decide medicine isn’t toxic when it really is
Probably Type II – people could be harmed

14. Identify the following decisions:
A government agency has received numerous complaints that a particular restaurant has been selling underweight hamburgers. The restaurant advertises that its patties are “a quarter pound” (4 ounces).
Identify the following decisions:
We decide the mean weight of hamburgers is less than 4 ounces when in fact they really are less than 4 ounces.
Correct decision!

15. Identify the following decisions:
A government agency has received numerous complaints that a particular restaurant has been selling underweight hamburgers. The restaurant advertises that its patties are “a quarter pound” (4 ounces).
Identify the following decisions:
We decide the mean weight of hamburgers is not less than 4 ounces when in fact they really are less than 4 ounces.
Type II Error

16. Identify the following decisions:
A government agency has received numerous complaints that a particular restaurant has been selling underweight hamburgers. The restaurant advertises that its patties are “a quarter pound” (4 ounces).
Identify the following decisions:
We decide the mean weight of hamburgers is less than 4 ounces when in fact they really are not less than 4 ounces.
Type I Error

17. How do we choose α? Assess consequences of type I and II errors
Pick largest α that is tolerable

18. ART is my BFF
Alpha: Reject True Ho
Beta: Fail to reject False Ho