Question

Type I and Type II Errors

.

Please discuss Type I and Type II errors.

What are they? Discuss their relationship with hypothesis testing.

Answer all parts of question!!! Do not plagiarize!! Write out the answer on here, don't post a picture of it! Answer must be long!

Answer 1

Type I error, also known as a “false positive”: the error of rejecting a null hypothesis when it is actually true. Plainly speaking, it occurs when we are observing a difference when in truth there is none. So the probability of making a type I error in a test with rejection region R is P (R | H₀ is true) . Alpha (α) is the probability that the test will lead to the rejection of the hypothesis tested when that hypothesis is true, ie, P(type I error)=α.

Type II error, also known as a "false negative" : the error of not rejecting a null hypothesis when the alternative hypothesis is the true state of nature. Plainly speaking, it occurs when we are failing to observe a difference when in truth there is one. So the probability of making a type II error in a test with rejection region R is P(R | H₁ is true) . Beta (β) is the probability that the test will reject the hypothesis tested when a specific alternative hypothesis is true, ie, P(probability of type II error)=β. power of the test is 1-β.

Hypothesis testing is the art of testing if variation between two sample distributions can just be explained through random chance or not. If we have to conclude that two distributions vary in a meaningful way, we must take enough precaution to see that the differences are not just through random chance. At the heart of Type I error is that we don't want to make an unwarranted hypothesis so we exercise a lot of care by minimizing the chance of its occurrence. Traditionally we try to set Type I error as .05 or .01 - as in there is only a 5 or 1 in 100 chance that the variation that we are seeing is due to chance. This is called the 'level of significance'. Again, there is no guarantee that 5 in 100 is rare enough so significance levels need to be chosen carefully. For example, a factory where a six sigma quality control system has been implemented requires that errors never add up to more than the probability of being six standard deviations away from the mean (an incredibly rare event). Type I error is generally reported as the p-value.

That is, just like a judge’s conclusion, an investigator’s conclusion may be wrong. Sometimes, by chance alone, a sample is not representative of the population. Thus the results in the sample do not reflect reality in the population, and the random error leads to an erroneous inference. A type I error (false-positive) occurs if an investigator rejects a null hypothesis that is actually true in the population; a type II error (false-negative) occurs if the investigator fails to reject a null hypothesis that is actually false in the population. Although type I and type II errors can never be avoided entirely, the investigator can reduce their likelihood by increasing the sample size (the larger the sample, the lesser is the likelihood that it will differ substantially from the population).

For an example : the null hypothesis is , H₀: The patient is not pregnant.

The first figure shows false positive that is type I error. The second figure shows the false negative that is type II error.