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All else equal, is specifying a smaller significance level in a hypothesis test likely to increase the probability of a Type I error and a Type II error?

A. Type I error: No ; Type II error: No
B. Type I error: No ; Type II error: Yes
C. Type I error: Yes ; Type II error: No

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Final answer:

Specifying a smaller significance level in a hypothesis test is likely to reduce the probability of a Type I error and increase the probability of a Type II error. This is because a smaller significance level means there is a more stringent criterion for rejecting the true null hypothesis, which makes it harder to detect a false null hypothesis.

Step-by-step explanation:

All else equal, specifying a smaller significance level, such as from 5% to 1%, in a hypothesis test reduces the probability of committing a Type I error but increases the probability of committing a Type II error. Therefore, the correct answer to the question about the likelihood of increasing the probability of Type I and Type II errors when reducing the significance level is: Type I error: No; Type II error: Yes.

A Type I error occurs when we reject a true null hypothesis, while a Type II error occurs when we fail to reject a false null hypothesis. As the significance level decreases, the criterion for rejecting the null hypothesis becomes more stringent, thus reducing the chance of incorrectly rejecting a true null hypothesis (Type I error). However, it also means that there is a higher chance we may not detect a false null hypothesis, thus increasing the risk of a Type II error.

Illustrative Example

Consider the hypothesis test where the null hypothesis (H₀) is that a new medication has no effect on a disease. If we set the significance level (α) to 0.01 instead of 0.05, we are saying we require stronger evidence to reject H₀. If the medication does have an effect (null hypothesis is false), but our sample data does not provide strong enough evidence at the 1% level, we may end up not rejecting a false H₀, hence committing a Type II error.

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