Errors in Hypothesis Tests

In hypothesis testing, when the p-value is less than α you reject H0. However, how often could a sample mean cause a p-value to be less than α, just by chance, when you really shouldn’t reject H0? α (ie. If α is 0.05 or 5%, there is a 5% chance, just by chance, that you reject the null hypothesis when you shouldn’t)

When you perform a hypothesis test you make a decision: reject H0 or fail to reject H0 When you make one of these decisions, there is a possibility that you could be wrong! That you made an error!

There are two decisions that we make; reject or fail to reject There are two decisions that we make; reject or fail to reject. Each could possibly be a wrong decision; therefore, there are two types of errors.

Type I error When you reject the null hypothesis that is really true Denoted by a Is the level of significance of the test a m0

Type II – fail to reject a false H0 Type II error When you fail to reject the null hypothesis when it is false Denoted by b Type I – reject a true H0 Fail to reject Reject The x-bar is really part of the H0 curve, but we mistake it as being part of the Ha curve The x-bar is really part of the Ha curve, but we mistake it as being part of the H0 curve a Type II – fail to reject a false H0 b

H0 True H0 False Reject Fail to reject Correct Correct Type I error Suppose H0 is false & we reject it, what type of decision was made? Type II error Correct Suppose H0 is false & we fail to reject it, what type of decision was made? Suppose H0 is true & we reject it, what type of decision was made? Suppose H0 is true & we fail to reject it, what type of decision was made? b

How do we word statements of type I & type II errors? “We say this decision when in reality this is true.” You replace the red, underlined words with words from context!

Consequences – are NOT the definitions of type I & II errors. They are what happens as a result of making that incorrect decision.

Consider a murder trial: Type I - that is why there must be evidence beyond a reasonable doubt! We don’t want to send innocent people to jail! What are H0: Ha: H0: defendant is innocent Ha: defendant is guilty What are the hypotheses? Type I error – Consequence: Type II error – What is a type I error? Say the defendant is guilty when really innocent What is a consequence of a type I error? An innocent person goes to prison Which of these errors does our society believe to be worse? What is a type II error? Say defendant is not guilty when really guilty A guilty person goes free What is a consequence of a type II error?

Facts: As a increases, b decreases As a decreases, b increases Every time you make a decision, you have potentially made an error. a & b are inversely related Fail to reject H0 Reject H0 a m0 b ma

Someone made an error with these products Facts continued: The seriousness of the error types is determined by the specific situations. Depending upon the situation type I or type II may be the more serious. We often DO NOT know if an error is made in real life. Except for cases like Firestone tires Drugs like: Phen-phen & Vioxx Someone made an error with these products

Sometimes, the seriousness depends upon the person’s point-of-view Lay’s Chip Company decides to accept a truckload of potatoes based upon results from a sample of potatoes from the truckload. 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 Say the potatoes are bad when they really are good Say the potatoes are good when they really are bad Sometimes, the seriousness depends upon the person’s point-of-view A type I error A type II error

What are the hypotheses? What are the 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. It has been determined that 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 regulations that prohibit a mean discharge above 150 degrees F, fifty water samples will be taken at randomly selected times, and the temperature of each sample recorded. What are the hypotheses? What are the Type I and II errors? Which is more serious? H0: m = 150 Ha: m >150 Type I : Say the temperature is above 150° when it’s really below. Most people would agree that the type II error would be more serious because it would endanger the river’s ecosystem. Type II: Say the temperature isn’t above 150° when it’s really above.

What are the hypotheses? What are the Type I & II errors? 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. What are the hypotheses? What are the Type I & II errors? Which is more serious? H0: medicine is not toxic Ha: medicine is toxic Type I: say medicine is toxic when it really isn’t Type II : say medicine isn’t toxic when it really is Most would consider a type II error more serious since people could be harmed.

How does one decide what a level to use? After assessing the consequences of type I and type II errors, identify the largest a that is tolerable for the problem. Use that a level for your level of significance.