Final answer:
To find the probability of making a Type II error (β) with the alternative mean time of 17 minutes, one would need to calculate the Z value with the formula Z = (X - μ) / (σ / √n) and then find the corresponding probability from the normal distribution. The provided significance level (0.01) seems to be for Type I error, not Type II, and without a Z value or a z-table, the exact β cannot be determined from the given options.
Step-by-step explanation:
To calculate the probability of making a Type II error, also known as β, under the alternative actual mean time of 17 minutes, we first need to determine the z-score for the sample mean under this alternative hypothesis. The z-score can be calculated using the formula
Z = (X - μ) / (σ / √n)
Substituting the given values we get,
Z = (17 - 15) / (4 / √35)
Once we find the Z value, we use the normal distribution to find the probability of observing a sample mean of 17 minutes or less when the true mean is indeed 17 minutes. This probability is the complement of the power of the test, which is 1 - β.
We are given the level of significance (α) as 0.01, which typically refers to the probability of making a Type I error, not Type II error. Therefore, without the Z value or a table to determine the cumulative probability, we cannot directly compute β from the information provided.
To provide an exact answer, we would need additional information or the ability to reference a z-table or use statistical software.
Hence, none of the provided options (1: 0.01, 2: 0.06, 3: 0.09, 4: 0.11) can be accurately chosen as the probability of making a Type II error purely based on the information given.