What do you call the probability of rejecting the null hypothesis when it is true?
Learning Outcomes
When you perform a hypothesis test, there are four possible outcomes depending on the actual truth (or falseness) of the null hypothesis H0 and the decision to reject or not. The outcomes are summarized in the following table:
The four possible outcomes in the table are: The decision is not to reject H0 when H0 is true (correct decision). The decision is to reject H0 when H0 is true (incorrect decision known as a Type I error). The decision is not to reject H0when, in fact, H0 is false (incorrect decision known as a Type II error). The decision is to reject H0 when H0 is false (correct decision whose probability is called the Power of the Test). Each of the errors occurs with a particular probability. The Greek letters α = probability of a Type I error = P(Type I error) = probability of rejecting the null hypothesis when the null hypothesis is true. β = probability of a Type II error = P(Type II error) = probability of not rejecting the null hypothesis when the null hypothesis is false. α and β should be as small as possible because they are probabilities of errors. They are rarely zero. The Power of the Test is 1 –β. Ideally, we want a high power that is as close to one as possible. Increasing the sample size can increase the Power of the Test. Suppose the null hypothesis, H0, is: Frank’s rock climbing equipment is safe.
α = probability that Frank thinks his rock climbing equipment may not be safe when, in fact, it really is safe. β = probability that Frank thinks his rock climbing equipment may be safe when, in fact, it is not safe. Notice that, in this case, the error with the greater consequence is the Type II error. (If Frank thinks his rock climbing equipment is safe, he will go ahead and use it.) try itSuppose the null hypothesis, H0, is: the blood cultures contain no traces of pathogen X. State the Type I and Type II errors.
Suppose the null hypothesis, H0, is: The victim of an automobile accident is alive when he arrives at the emergency room of a hospital.
α = probability that the emergency crew thinks the victim is dead when, in fact, he is really alive = P(Type I error). β = probability that the emergency crew does not know if the victim is alive when, in fact, the victim is dead =P(Type II error). The error with the greater consequence is the Type I error. (If the emergency crew thinks the victim is dead, they will not treat him.) try itSuppose the null hypothesis, H0, is: a patient is not sick. Which type of error has the greater consequence, Type I or Type II? The error with the greater consequence is the Type II error: the patient will be thought well when, in fact, he is sick, so he will not get treatment. It’s a Boy Genetic Labs claim to be able to increase the likelihood that a pregnancy will result in a boy being born. Statisticians want to test the claim. Suppose that the null hypothesis, H0, is: It’s a Boy Genetic Labs has no effect on gender outcome.
The error of greater consequence would be the Type I error since couples would use the It’s a Boy Genetic Labs product in hopes of increasing the chances of having a boy. try it“Red tide” is a bloom of poison-producing algae–a few different species of a class of plankton called dinoflagellates. When the weather and water conditions cause these blooms, shellfish such as clams living in the area develop dangerous levels of a paralysis-inducing toxin. In Massachusetts, the Division of Marine Fisheries (DMF) monitors levels of the toxin in shellfish by regular sampling of shellfish along the coastline. If the mean level of toxin in clams exceeds 800 μg (micrograms) of toxin per kg of clam meat in any area, clam harvesting is banned there until the bloom is over and levels of toxin in clams subside. Describe both a Type I and a Type II error in this context, and state which error has the greater consequence. In this scenario, an appropriate null hypothesis would be H0: the mean level of toxins is at most 800 μg, H0 : μ0 ≤ 800 μg.
In summary, the more dangerous error would be to commit a Type II error, because this error involves the availability of tainted clams for consumption. A certain experimental drug claims a cure rate of at least 75% for males with prostate cancer. Describe both the Type I and Type II errors in context. Which error is the more serious?
In this scenario, the Type II error contains the more severe consequence. If a patient believes the drug works at least 75% of the time, this most likely will influence the patient’s (and doctor’s) choice about whether to use the drug as a treatment option. try itDetermine both Type I and Type II errors for the following scenario: Assume a null hypothesis, H0, that states the percentage of adults with jobs is at least 88%. Identify the Type I and Type II errors from these four statements. a)Not to reject the null hypothesis that the percentage of adults who have jobs is at least 88% when that percentage is actually less than 88% b)Not to reject the null hypothesis that the percentage of adults who have jobs is at least 88% when the percentage is actually at least 88%. c)Reject the null hypothesis that the percentage of adults who have jobs is at least 88% when the percentage is actually at least 88%. d)Reject the null hypothesis that the percentage of adults who have jobs is at least 88% when that percentage is actually less than 88%. Type I error: c Type I error: b Concept ReviewIn every hypothesis test, the outcomes are dependent on a correct interpretation of the data. Incorrect calculations or misunderstood summary statistics can yield errors that affect the results. A Type I error occurs when a true null hypothesis is rejected. A Type II error occurs when a false null hypothesis is not rejected. The probabilities of these errors are denoted by the Greek letters α and β, for a Type I and a Type II error respectively. The power of the test, 1 – β, quantifies the likelihood that a test will yield the correct result of a true alternative hypothesis being accepted. A high power is desirable. Formula Reviewα = probability of a Type I error = P(Type I error) = probability of rejecting the null hypothesis when the null hypothesis is true. β = probability of a Type II error = P(Type II error) = probability of not rejecting the null hypothesis when the null hypothesis is false. |