Final answer:
The probability that the sample mean is more than 4 for a binomial distribution X ~ B(9, 0.5) with a sample size of 60 can be found using the Central Limit Theorem to approximate the distribution to a normal distribution. The mean, standard error, and z-score would be used to find this probability, but the exact answer cannot be determined without the normal probability calculations.
Step-by-step explanation:
The question is asking for the probability that the sample mean of a binomial distribution is more than 4, given a random sample size of 60. To find this probability, we need to understand the distribution of the sample mean, which in this case, due to the Central Limit Theorem, can be approximated by a normal distribution if the sample size is large enough.
In this scenario, the binomial distribution in question is X ~ B(9, 0.5), and we're evaluating a sample of size 60. The mean (μ) of the binomial distribution can be found using the formula μ = n*p, where n is the number of trials and p is the probability of success. Thus, μ = 9*0.5 = 4.5. However, since we're looking for the sample mean of size 60, we should use the Central Limit Theorem to approximate the binomial distribution to a normal distribution because the sample size is large. The standard error (SE) for the sample mean can be calculated using the formula SE = √(p*q/n), where q = 1 - p. With this approximation, we'd use a normal distribution table or software to find the probability that the sample mean is more than 4.
The provided options imply that the calculation has already been made, so we would select the option that corresponds to the correct normal probability. However, without the actual normal probability calculations presented in the question, we cannot definitively choose an answer from the given options a) 0.022, b) 0.0017, c) 0.9983, or d) 0.977.