Final answer:
A Type I error is rejecting a true null hypothesis, and the minimum sample mean with a 5% probability of incorrectly rejecting the null hypothesis would be the critical value at the 0.05 alpha level for the distribution relevant to the test.
Step-by-step explanation:
When conducting a hypothesis test, a Type I error occurs when the null hypothesis is rejected even though it is actually true. Based on the information provided, the Type I error in this case would be rejecting the assumption that the mean is equal to or less than a particular value (for example, μ = 65 as stated in the given hypothesis test scenario) when, in reality, the mean is indeed equal to or less than that value.
To determine the minimum value of the sample mean for which there is a 5% probability of rejecting the null hypothesis even if it were true, we would typically look at the corresponding critical value from a statistical table or software that matches the alpha level of 0.05 in the context of the specific distribution relevant to the test (e.g., t-distribution, z-distribution). This value is often referred to as the cutoff score or critical value. If the sample mean is greater than this critical value, then the null hypothesis would be rejected at the 5% significance level.