Errors in Hypothesis Tests

When you perform a hypothesis test you make a decision: When you make one of these decisions, there is a possibility that you could be wrong! That you made an error! reject H 0 or fail to reject H 0

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  –Is the level of significance of the test  

Fail to rejectReject Type II error When you fail to reject the null hypothesis when it is false Denoted by    Type I – reject a true H 0 Type II – fail to reject a false H 0 The sample statistic (p- hat) is really part of the H a curve, but we mistake it as being part of the H 0 curve The sample statistic (p- hat) is really part of the H 0 curve, but we mistake it as being part of the Ha curve

H 0 True H 0 False Reject Fail to reject Type I error Correct Type II error Suppose H 0 is true & we fail to reject it, what type of decision was made? Suppose H 0 is false & we reject it, what type of decision was made? Suppose H 0 is true & we reject it, what type of decision was made? Suppose H 0 is false & we fail to reject it, what type of decision was made?  

How do we word statements of type I & type II errors? “We decide 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: What are the hypotheses? Type I error – Consequence: Type II error – Consequence: 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 H 0 : defendant is innocent H a : defendant is guilty Which of these errors does our society believe to be worse? 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 H 0 : H a : What is a type I error? What is a type II error? What is a consequence of a type I error? What is a consequence of a type II error?

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? Decide the potatoes are bad when they really are good Decide the potatoes are good when they really are bad H 0 : potatoes good H a : potatoes bad A type I error A type II error Sometimes, the seriousness depends upon the person’s point-of-view

Errors Continued

Facts: Every time you make a decision, you have potentially made an error.  &  are inversely related 00  aa  Fail to reject H 0 Reject H 0 As  decreases,  increases As  increases,  decreases

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. DO NOTWe 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

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 following decisions: We decide the true proportion of vaccinated people who do not get the flu is less than 70% when in fact it really is less. Correct decision!

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 following decisions: We decide the true proportion of vaccinated people who do not get the flu is not less than 70% when in fact it really is less. Type II error

Most people would agree that the type II error would be more serious because it would endanger the river’s ecosystem. 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? H 0 :  = 150 H a :  >150 Type I : Decide the temperature is above 150° when it’s really below. Type II: Decide the temperature isn’t above 150° when it’s really above.

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? H 0 : medicine is not toxic H a : 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 Most would consider a type II error more serious since people could be harmed.

ON YOUR OWN: Suppose a blood test was used to detect a serious illness (like HIV). State the hypotheses. State Type I & Type II errors and their consequences. Which is more serious?

Consider a blood screening test for certain diseases, such as HIV. What are the hypotheses? –H 0 : person is well –H a : person is sick Type I error: –The blood tested positive for disease when person is really well. (called a false positive) Type II error: –The blood tested negative for disease when person is really sick. (called a false negative)

Which is more serious (which should be made small)? –Type I – so with a screening blood test, we expect more false positives. That is why doctors will say we need to do more testing. The second test is more accurate & more expensive to run. Now, consider the more definitive, and more expensive, blood test. Which would you want to make smaller? –Now the type I error should be smaller so that only the people who really have the disease tests positive.

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