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## Statistics and probability

### Course: Statistics and probability>Unit 12

Lesson 2: Error probabilities and power

# Type 1 errors

A Type 1 error occurs when the null hypothesis is true, but we reject it because of an usual sample result. Created by Sal Khan.

## Want to join the conversation?

• So what is the exact relationship between Type 1 error and the p-value?
(1 vote) • This might sound confusing but here it goes: The p-value is the probability of observing data as extreme as (or more extreme than) your actual observed data, assuming that the Null hypothesis is true.

A Type 1 Error is a false positive -- i.e. you falsely reject the (true) null hypothesis. In addition, statisticians use the greek letter alpha to indicate the probability of a Type 1 error.

So, as you can see, the two terms are related but not "exactly related" since there is some slight difference in their definitions...

Anyway, inn practice, you would determine your level of significance beforehand (your alpha) and then reject the null hypothesis if your p-value turns out to be smaller than alpha.

• Which error does "power" correspond to? And what does that mean? • The power of a test is 1-(type 2 error). Keeping in mind that type 2 error is the probability of failing to reject H0 given that H1 is true. So the power of a test tells us something about how strong the test is, that is how well the test can differentiate between H0 and H1. To improve the power of a test one can lower the variance or one can increase alfa (type 1 error).

Power curves shows the power of the test given different values of H1. The longer H1 is from H0 the easier it is to differen
• How do I calculate power? Some of the other answers mention beta but don't say how to calculate either beta or power. • can I see an example of
the type 1 error being worked out from using a worded statement? • Here's an easy example of a Type I error. Suppose you are tested for an extremely rare disease that affects only 1 in a million people. The test is 99.9% accurate. Your test comes back positive. It would almost certainly be a Type I error to conclude you have that disease. Here's why.

0.1% of the time, the test produces the wrong answer. Thus, out of a million people, you would get 1000 false positives. You would expect to get 1 person with the disease that has positive test.

Thus, you would reasonably expect 1000 of the 1001 positives to be false positives. This makes it nearly certain that you don't have the disease.

Thus, you almost certainly made a Type I error if you assumed you had the disease. The null hypothesis is that you don't have the disease and you almost certainly falsely rejected that hypothesis.

Therefore, the accuracy of a test must be in keeping with how likely it is that the hypothesis is true. A condition that affects half the population can reasonably be tested with a procedure that is 99.9% accurate. But a rare disease requires much greater accuracy because the false positives would tend to be far more common than true positives without extreme accuracy.

[Actually, I simplified the math a bit, you should be even more dubious about test than I indicated because a sample of a million people is not large enough to reasonably expect 1 person to have the disease. With 1 million people, you'd only have a 63% chance of someone having the disease, you would need about 2.5 million people to have a greater than 90% chance of having someone with the disease.]
• Where exactly is the null hypothesis explained? I wish there was some sort of logical flow to these videos... • I think I understand everything else, but where does the 0.5% come from? why that exact value? • I assume you mean the 0.05 (or 5%). That's basically just an arbitrary number. A famous Statistician by the name of R.A. Fisher commented that about 1 in 20 - which is 0.05 - is a convenient value to use. He used and apparently recommended that when writing some books. I guess it caught on and stuck.

There's nothing special about 0.05. Some professions aren't that strict, some are far more strict. If I recall correctly, physicists use something like 0.0001. And people who sort through dozens or hundreds of genes to try and find out what genes may influence some condition will use something even smaller.
• I don't understand, first of all, what is a null hypothesis? • What's the difference between critical value and significance level ? OR are they same?   