Final answer:
The statements concern the outcomes of hypothesis testing at different levels of significance. The first and third statements are true due to the nature of alpha levels, while the second statement is false as not rejecting at a lower alpha doesn't imply the same at a higher alpha.
Step-by-step explanation:
The question revolves around the concept of hypothesis testing in statistics, especially the significance levels (α) and the potential outcomes when making decisions about the null hypothesis (H0). Here are the statements evaluated:
- (a) If H0 is rejected at α = 0.01, it must also be rejected at α = 0.05. This statement is true because a lower alpha level (0.01) means a more stringent criterion for rejection. Therefore, if H0 is rejected at a more stringent level, it would certainly be rejected at a more lenient level (0.05).
- (b) If H0 is not rejected at α = 0.01, it must also not be rejected at α = 0.05. This is false because not rejecting the null at a stricter significance level does not guarantee the same decision at a less strict level. The p-value might be between 0.01 and 0.05, which would lead to not rejecting H0 at 0.01 but rejecting it at 0.05.
- (c) If H0 is not rejected at α = 0.05, it must also not be rejected at α = 0.01. This statement is true because a higher alpha level (0.05) is less stringent. Thus, if the null hypothesis is not rejected at this level, it will also not be rejected at the more stringent level of 0.01.
Understanding the probability of type 1 (rejecting H0 when it is true) and type 2 errors (not rejecting H0 when it is false) is essential in hypothesis testing. The significance level, p-value, and evidence from the data help determine whether to reject or fail to reject the null hypothesis.