To understand what happens when we conclude that the mean is greater than 53 when its true value is actually \( \mu \leq 53 \), let's break down the hypothesis testing process and the types of errors involved.
1. Hypotheses:
- Null Hypothesis (\(H_0\)): \(\mu \leq 53\)
- Alternative Hypothesis (\(H_1\)): \(\mu > 53\)
2. Decision Making:
- We can either reject the null hypothesis (\(H_0: \mu \leq 53\)) or fail to reject it.
- If we reject \(H_0\), we are supporting \(H_1\), which means we believe the mean is greater than 53 (\(\mu > 53\)).
3. Types of Errors:
- Type I Error (False Positive): Rejecting the null hypothesis when it is actually true.
- Type II Error (False Negative): Failing to reject the null hypothesis when the alternative hypothesis is actually true.
Given the scenario in the question:
- We conclude that the mean \(\mu\) is greater than 53.
- If the true value is \(\mu \leq 53\), this means our conclusion is incorrect because we rejected the null hypothesis \(H_0: \mu \leq 53\) when it is actually true.
This situation corresponds to a Type I Error. Thus, the correct choice is:
b. We have made a Type I error
This means we've incorrectly rejected the null hypothesis when it was actually true.
