Final answer:
The probability of committing a Type II error in this hypothesis test, given the true mean is μ = 50, is approximately 0.9772.
Step-by-step explanation:
To calculate the probability of committing a Type II error (also known as β or 'beta'), we need to find the probability that the test will fail to reject the null hypothesis H0: μ = 55 when in fact the true mean is μ = 50. In this scenario, the test is left-tailed because the alternative hypothesis is H1: μ < 55.
Given the standard deviation (σ) of 20 and a sample size (n) of 64, we can calculate the standard error of the mean, which is σ/√n = 20/8 = 2.5. To find the z-score that corresponds to α = 0.10, we look up the z-table and find a value of approximately -1.28. This z-score is the cut-off point for our test statistic.
If the true mean is 50, we calculate the z-score using this mean. The z-score for the true mean is (50 - 55)/(2.5) = -2. The probability corresponding to this z-score is the probability of not rejecting H0, which is the probability of a Type II error. We find this probability in the z-table, which corresponds to the area to the right of a z-score of -2. This area is approximately 0.9772.
Hence, the probability of committing a Type II error, given μ is 50, is approximately 0.9772.