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Types of Error Power of a Hypothesis Test AP Statistics - Chapter 21 We make decisions based on a probability … but what if … we’re WRONG?!? When we perform a hypothesis test: In our hypothesis... In real life... Reject HO Fail to reject HO HO true HO false Type I α Error ✓ ✓ Type II β Error When we perform a hypothesis test: In our hypothesis... In real life... In medical testing, weHOcall this a HO “false true positive” false Reject HO to reject “falseFail negative” HO Type I α Error ✓ ✓ Type II β Error Consider a test for a serious disease... What are the hypotheses? Ho: the person is healthy Ha: the person is NOT healthy/has the disease Type I Error? We decide the person has the disease … when in reality, they’re healthy. Consequence? We scare the person for no reason. They unnecessarily go in for further testing or treatment.. Consider a test for a serious disease... What are the hypotheses? Ho: the person is healthy Ha: the person is NOT healthy/has the disease Type II Error? We say there’s not enough evidence to conclude the person has the disease (in other words, we think they’re healthy), when in reality, they actually do have it. Consequence? The person fails to get treatment for a disease they have.. Lay’s chip company tests a sample of potatoes from a truckload for E-coli to determine whether or not to Ho: the potatoes are good accept the truckload. Ha: the potatoes have E-coli Type I Error? Type II Error? We decide the potatoes have E-coli, when they really don’t. We decide the potatoes are good, when they actually have E-coli. Which error is worse? Which error is more concerning? What if you’re the potato farmer? What if you’re the CEO of Lay’s chip company? What if you’re a person buying potato chips? When we perform a hypothesis test: In our hypothesis... If we decrease P(Type I error), the probability of Type II error increases by default. Statisticians have to decide which error they want to avoid more, Reject knowing that decreasing one will increase the other! H O Fail to reject HO In real life... HO true Type I Error ✓ HO false α ✓ Type II β Error When we perform a hypothesis test: In our hypothesis... In real life... HO true Reject HO And it works the other way around as well: if we decrease P(Type II error), the probability of Type I error Fail to reject increases by default. HO Type I α Error ✓ HO false ✓ Type II Error β A school district is considering purchasing laptops for all of high school students, in hopes that using the devices will improve achievement on end-of-year exams. What are the hypotheses? Ho: student achievement stays the same/does not improve Ha: student achievement improves Is this a one-tailed or two-tailed test? One-tailed. The district wants to show an improvement in achievement. (They’re not going to test for just a change in achievement, because they wouldn’t buy laptops if achievement got worse!) A school district is considering purchasing laptops for all of high school students, in hopes that using the devices will improve achievement on end-of-year exams. What are the hypotheses? Type I Error? Type II Error? Ho: student achievement stays the same/does not improve The district decides that laptops The district decides that laptops do improve student achievement, don’tHa: change student achievement, student achievement improves when they actually don’t. when they actually do. Is this a one-tailed or two-tailed test? Consequence? Consequence? District spends $$$ onThe laptops Students don’t get access to One-tailed. district wants to show an improvement in achievement. that aren’t actually helping laptops thatbecause wouldthey actually (They’re not going to test for just a change in achievement, wouldn’t buy got worse!) anything. helplaptops them ifbeachievement more successful. Ho: the defendant is innocent Ha: the defendant is guilty (let’s think about a criminal trial) Let’s pretend we have a defendant that we K N O W is guilty of a crime. What do we NEED in order to convict the criminal? (in other words, to reject the null hypothesis?) STRO NG evide nce! Without it, we risk committing a Typ e II error. Or … what if I’ve developed a new medication that I KNOW is better than the previous one? What do I need if I want to “prove” that the new medication is better? STRONG evidence! Or … what if I’ve developed a new medication that I What do I need if I want to “prove” STRONG In hypothesis testing, that that the newevidence! medication is better? KNOW “evidence” we need is called is better than the POWER. previous one? POWER the probability of rejecting Ho, when Ho is false. (or, the probability of concluding Ha is true, when Ha IS true!) POWER = 1-β In our hypothesis... When we perform a hypothesis test: PO W Reject HO Fail to reject HO In real life... HO true HO false Type I α Error ✓ ✓ Type II β Error ER ! When we have a Ha that we KNOW is true, but still need the hypothesis test to PROVE it’s true we need POWER to be as big as it can possibly be. Here are 3 ways to increase power: 1. Increase α - Increasing alpha lowers P(Type II) or β, thus increasing power. But … this will also increase P(Type I), which may not be ideal. 2. Increase n - Increasing sample size will lower both P(Type I) & P(Type II), and increase power. But … taking a bigger sample size is not always possible or realistic. 3. Increase effect size - This means … make the new thing REALLY better, by a lot. This would lower P(Type I) and P(Type II) and increase power. It’s just not something statisticians usually have any control over.
