As the alternative mean approaches the hypothesized mean, what can we say about the risk?
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A. B. C. D. E.Explanation
When the alternative mean approaches the hypothesized mean, there is a greater possibility of accepting the null when it is actually false since they are close together.
The question is asking about the relationship between the alternative mean and the hypothesized mean, and its impact on the risk of Type I and Type II errors.
Type I and Type II errors are statistical errors that can occur in hypothesis testing.
In hypothesis testing, we set a significance level (usually denoted as α) to determine the threshold for rejecting the null hypothesis. The significance level represents the maximum acceptable probability of committing a Type I error.
Now, let's consider the statement "As the alternative mean approaches the hypothesized mean."
When the alternative mean approaches the hypothesized mean, it means that the difference between the two means becomes smaller. In this case, the alternative hypothesis (which assumes a difference between means) becomes less likely.
Based on this understanding, we can analyze the answer choices:
A. Smaller risk of a Type II error: This statement implies that as the alternative mean approaches the hypothesized mean, the risk of failing to reject a false null hypothesis (Type II error) decreases. However, this is incorrect because a smaller difference between the alternative mean and the hypothesized mean would make it more difficult to detect a true difference, increasing the risk of Type II error.
B. Greater risk of a Type II error: This statement correctly explains that as the alternative mean approaches the hypothesized mean, the risk of Type II error increases. This is because the smaller difference between the means reduces the ability to detect a true difference, increasing the likelihood of failing to reject a false null hypothesis.
C. Smaller risk of a Type I error: This statement implies that as the alternative mean approaches the hypothesized mean, the risk of rejecting a true null hypothesis (Type I error) decreases. However, this is incorrect because the risk of Type I error is determined by the significance level (α), not by the proximity of the alternative mean to the hypothesized mean.
D. None of these answers: This answer choice does not provide any explanation and is not correct based on the analysis provided.
E. Greater risk of a Type I error: This statement suggests that as the alternative mean approaches the hypothesized mean, the risk of rejecting a true null hypothesis (Type I error) increases. However, this is incorrect because, as mentioned earlier, the risk of Type I error is determined by the significance level (α) and is not directly influenced by the proximity of the means.
In conclusion, the correct answer is B. As the alternative mean approaches the hypothesized mean, there is a greater risk of committing a Type II error, not Type I error.