Step 1 :The problem asks whether the sample size is large enough to use the Central Limit Theorem for means. The Central Limit Theorem states that if the sample size is large enough (usually n > 30), the sampling distribution of the mean will be approximately normally distributed, regardless of the shape of the population distribution. In this case, the sample size is 100, which is greater than 30. Therefore, the sample size is large enough to use the Central Limit Theorem for means. The final answer for this part is \(\boxed{\text{D. Yes, it is large enough because the sample size of 100 is greater than 25.}}\)
Step 2 :The problem also asks for the mean and standard error of the sampling distribution. The mean of the sampling distribution is equal to the population mean, which is \$77000. The standard error of the sampling distribution is equal to the population standard deviation divided by the square root of the sample size. In this case, the standard error is \$32000 divided by the square root of 100, which is \$3200. Therefore, the mean is \$77000 and the standard error is \$3200.
Step 3 :The problem also asks for the probability that the sample mean will be more than \$3200 away from the population mean. This is equivalent to finding the probability that the sample mean is either less than \$77000 - \$3200 = \$73800 or more than \$77000 + \$3200 = \$80200. To find this probability, we would need to use the standard normal distribution table or a calculator with a normal distribution function. However, the problem does not provide enough information to calculate this probability.