Needing some help to clarify this problem and help finding the right answer.
_b___ 2. Which of the following accurately describes the critical region?
a. outcomes with a very low probability if the null hypothesis is true
b. outcomes with a high probability if the null hypothesis is true
c. outcomes with a very low probability whether or not the null hypothesis is true
d. outcomes with a high probability whether or not the null hypothesis is true
The critical region CR, or rejection region RR, is a set of values of the test statistic for which the null hypothesis is rejected in a hypothesis test. That is, the sample space for the test statistic is partitioned into two regions; one region (the critical region) will lead us to reject the null hypothesis H0, the other will not. So, if the observed value of the test statistic is a member of the critical region, we conclude "Reject H0"; if it is not a member of the critical region then we conclude "Do not reject H0".
To determine the right answer, let's start by understanding the critical region in hypothesis testing.
In hypothesis testing, we start with a null hypothesis (H₀), which is the statement we want to test, and an alternative hypothesis (H₁), which is the statement we suspect might be true. The critical region is a range of values or outcomes that we define in advance, which will lead us to reject the null hypothesis if the test statistic falls within this range.
Now, let's examine the answer choices:
a. outcomes with a very low probability if the null hypothesis is true - This is correct. The critical region consists of outcomes that have a low probability of occurring if the null hypothesis is true. In other words, if we observe a result within the critical region, it suggests that the null hypothesis is unlikely to be true.
b. outcomes with a high probability if the null hypothesis is true - This is incorrect. The critical region consists of outcomes with a low probability if the null hypothesis is true, not a high probability.
c. outcomes with a very low probability whether or not the null hypothesis is true - This is incorrect. The critical region is designed to specifically consider the probability when the null hypothesis is true, so it is not about whether or not the null hypothesis is true.
d. outcomes with a high probability whether or not the null hypothesis is true - This is incorrect. The critical region consists of outcomes with a low probability if the null hypothesis is true, not a high probability.
Therefore, the correct answer is (a) outcomes with a very low probability if the null hypothesis is true.