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Wize AP Statistics Textbook > Hypothesis Testing for Mean and Proportion

Type I and Type II Errors

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Type I and Type II Errors

When we conduct hypothesis tests, two things could go wrong:

Type I Error: You reject Ho when Ho is true (“false positive”)
e.g. The doctor concludes that the patient is well and could be discharged from the hospital – but in fact he is not well!

Type II Error: You do not reject Ho when Ho is false (“false negative”)
e.g. Serena concludes that texting while driving does not increase car accidents – but it does!

Summary



Watch Out!
Our conclusion is based on our sample data; we may never really know if HoH_o is in fact true or false.

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Example

HoH_o: “The old bridge is not dangerous.”
HaH_a: “The old bridge is dangerous.”


In this example, committing which type of error is more serious: Type I or Type II?

Type I error: Reject Ho →H_o\ \rightarrow you believe that the old bridge is dangerous – but it is not!

Type II error: Fail to reject Ho →H_o\ \rightarrow you believe that the old bridge is not dangerous – but it is!

In this example, Type II error is more serious because you end up using a bridge that is dangerous. A Type I error is when you decide not to use the bridge even though it is safe, which is just an inconvenience.



Wize Tip
If you reject Ho →\rightarrowType I Error is possible (Type II Error is not possible)
If you fail to reject Ho →\rightarrowType II Error is possible (Type I Error is not possible)


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Consider this conclusion: “We fail to reject HoH_o and we proved that the old bridge is not dangerous.”

THIS IS WRONG! If we do not reject HoH_o, that doesn’t mean we proved it’s true that the bridge is not dangerous. We just do not have evidence that the bridge is dangerous (HaH_a).


Watch Out!
Failing to reject HoH_o that does not confirm or prove that HoH_o is true!

Correct conclusion: “We do not reject HoH_o and conclude that there is no evidence that the bridge is dangerous.”

Summary

  • When we reject HoH_o, we have evidence to support HaH_a.
  • If HoH_o is in fact true but we reject it, then we’ve committed a Type I Error.
  • When we do not reject HoH_o, we have no evidence to support HaH_a but it does not prove that HoH_otrue; we just didn’t find evidence against HoH_o
  • If HoH_o is in fact false but we failed to reject it, then we’ve committed a Type II Error.

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Example: Type I and Type II Errors


Let’s use a law example. Suppose it is true that Morgan stole your iPad (but you do not know that). You and the prosecutor put her on trial:

HoH_o: Morgan did not steal your iPad (Not Guilty)
HaH_a: Morgan stole your iPad (Guilty)

Morgan pleads not guilty. Sadly, you found no evidence that she is guilty (HaH_a) so you cannot prosecute her.

(a) Did we prove that Morgan is not guilty?

No, that does not prove that she is truly not guilty (HoH_o). You just have no evidence to conclude that she is guilty (HaH_a).

(b) Did we commit a Type I error, Type II error, or make the correct decision?

Since you lack evidence, you fail to reject HoH_o so you conclude that there is no evidence that Morgan is guilty. But remember: HoH_o is false because she did steal your iPad (and got away with it)! You failed to reject HoH_o, and, because HoH_o is false, we have committed a Type II Error.

Practice: Type I and Type II Errors


The mayor believes that the crime rate has stayed the same but in fact it is much worse.

Determine if a Type I Error, Type II Error, or correct decision has been made.

Practice: Type I and Type II Errors


Carbo Max claimed to have developed a pill that will allow people to eat carbs and not gain weight. People who tried it gained weight. Lawsuits ensued.

Determine if a Type I Error, Type II Error, or correct decision has been made.

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Power

Making the Correct Decision

After running a hypothesis test, you want to draw the right conclusion. If the null hypothesis is in fact false, you should reject it. Power is the probability of correctly rejecting a false null hypothesis.
  • You want a high power.
  • If theP(reject Ho∣Ho false)P(reject \ H_o|H_o \ false) close to 1, it means that, given that the null hypothesis is false, there is a high probability that you will reject it.
Example

Suppose you have a borderline p-value of 0.054. Our decision to reject HoH_o or not depends on the significance level, α\alpha :


Notice that we are very close to rejecting HoH_o at α=0.05\alpha=0.05. If we reject HoH_o at α=0.05\alpha=0.05 (when we should not reject HoH_o), then we commit a Type I Error. Therefore, the probability of committing a Type I Error is α\alpha:


P(Reject Ho∣Ho true)=P(Type I Error)=αP(Reject \ H_o| H_o \ true) = P(Type \ I \ Error)= \alpha →\rightarrow significance level


Wize Concept
If you are afraid of making a Type I Error, then you should lower α\alpha.


Unfortunately, if you reduce your chance of making a Type I Error – guess what? You’ll increase your chance of making a Type II Error!

The probability of committing a Type II Error is β\beta :

P(Fail to reject Ho∣Ho false)=P(Type II Error)=βP(Fail\ to \ reject \ H_o| H_o \ false) = P(Type \ II \ Error)= \beta

Conversely:

P(Reject Ho∣Hofalse)=1−βP(Reject\ H_o| H_o false) = 1-\beta →\rightarrow power


Wize Concept
Power is the probability of correctly rejecting a false null hypothesis.


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Example

Ho:μ=10H_o:\mu=10
Ho:μ>10H_o:\mu>10

  • Assuming HoH_o is true.
  • Suppose the true mean is in fact greater than 10!
  • Therefore, HoH_o is false.



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Is Type I Error or Type II More Dangerous?

We should strive to optimally keep the probability of each type of error small by selecting a reasonably small α\alpha (i.e. α=0.05\alpha=0.05 or lower) and a high power.

You should also take into consideration which type of error is more serious:
  • If it is more dangerous to commit a Type I error, then α\alpha should be small.
  • If it is more dangerous to commit a Type II error, then β\beta should be small.
What is the power of a hypothesis test?