In hypothesis testing, β is the probability of committing an error of Type II. The power of the test, 1 - β, is then:
A. the probability of failing to reject H0.
B. the probability of rejecting H0 when HA is true
C. the probability of failing to reject H0 when H0 is true
D.. the probability of failing to reject H0 when HA is true
E. the probability of rejecting H0 when H0 is true
B. The probability of rejecting H0 when HA is true.
Oh, hypothesis testing! It sounds like a very serious business, but don't worry - Clown Bot is here to lighten things up!
Let's tackle this question with a touch of humor, shall we? Imagine you're throwing a party and you invite all your friends. H0 is the hypothesis that nobody will show up to your party, while HA is the hypothesis that at least one person will show up.
Now, β, the probability of committing a Type II error, is like the chance of mistakenly thinking nobody shows up when in fact there is someone hiding in the corner, munching on snacks. So, think of β as your very cautious friend who always thinks the worst-case scenario is true.
On the other hand, the power of the test, 1 - β, is like the opposite of your cautious friend. It's the probability of rejecting H0 when HA is true. It's saying, "Hey party people, come on in! I know there's someone out there!"
So, to answer the question, the power of the test, 1 - β, is option B. It's the probability of rejecting H0 when HA is true. It's like the excitement and joy you feel when your party is a success and people actually show up!
Now go out there and throw the best statistical party ever! 🎉
The correct answer is B. the probability of rejecting H0 when HA is true.
The power of a statistical test is the probability of correctly rejecting the null hypothesis (H0) when the alternative hypothesis (HA) is true. In other words, it is the probability of detecting a true effect or difference if it really exists. So, 1 - β (where β is the probability of committing an error of Type II) gives us the power of the test.
To determine the correct answer, let's break down the definitions of the terms involved:
β (beta) is the probability of committing an error of Type II. Type II error occurs when the null hypothesis (H0) is false, but we fail to reject it.
The power of the test, 1 - β, is the probability of correctly rejecting the null hypothesis when the alternative hypothesis (HA) is true. In other words, it is the probability of correctly detecting a true effect or finding evidence against the null hypothesis.
Now let's match these definitions with the answer options:
A. The probability of failing to reject H0.
This option does not match the definition of power, which is about correctly rejecting the null hypothesis when it is false. It refers to not rejecting the null hypothesis when it is true.
B. The probability of rejecting H0 when HA is true.
This option aligns with the definition of power. If the alternative hypothesis (HA) is true, and the test correctly rejects the null hypothesis (H0), then this is the power of the test.
C. The probability of failing to reject H0 when H0 is true.
This option refers to the correct outcome when the null hypothesis is true, meaning not rejecting it. It does not match the definition of power, which is about correctly detecting the effect when it is present.
D. The probability of failing to reject H0 when HA is true.
This option is the same as option C, which refers to not rejecting the null hypothesis when the alternative hypothesis is actually true. It is not the power of the test.
E. The probability of rejecting H0 when H0 is true.
This option refers to rejecting the null hypothesis when it is true. It does not match the definition of power, which relates to correctly rejecting H0 when HA is true.
Based on the definitions and matching them with the options, the correct answer is B. The power of the test is the probability of rejecting the null hypothesis when the alternative hypothesis is true.