Based on U.S. Bureau of Labor Statistics, on average, electricians earn approximately $\$ 60,040$ per year or $\mu$ $=\$ 29$ dollars per hour in the United States in 2021. Assume that the population standard deviation of earning per hour is $\sigma=12$ dollars. A sample of 40 electricians is randomly selected.
i. Describe the probability distribution of the sample mean of hourly earnings.
ii. What are the mean and the standard deviation for the sampling distribution of the sample mean?
iii. What is the probability that the average hourly earnings of the forty randomly selected electricians is between $\$ 25$ and $\$ 30$ ?

Step 1 ：The problem is asking for the probability distribution of the sample mean of hourly earnings, the mean and standard deviation for the sampling distribution of the sample mean, and the probability that the average hourly earnings of the forty randomly selected electricians is between $25 and $30.

Step 2 ：Since we are dealing with a sample mean and a known population standard deviation, we can use the Central Limit Theorem to say that the distribution of the sample mean will be approximately normally distributed.

Step 3 ：The mean of the sampling distribution (also known as the expected value of the sample mean) is equal to the population mean. The standard deviation of the sampling distribution (also known as the standard error) is equal to the population standard deviation divided by the square root of the sample size.

Step 4 ：To find the probability that the sample mean is between $25 and $30, we can use the standard normal distribution (Z-distribution) and the standard error. We will need to find the Z-scores for $25 and $30, and then find the area between these two Z-scores on the standard normal distribution.

Step 5 ：Given that the population mean (mu) is $29, the population standard deviation (sigma) is $12, and the sample size (n) is 40, we can calculate the mean of the sampling distribution to be $29 and the standard deviation of the sampling distribution to be approximately $1.90.

Step 6 ：We can then calculate the Z-scores for $25 and $30 to be approximately -2.11 and 0.53 respectively.

Step 7 ：Finally, we can calculate the probability that the average hourly earnings of the forty randomly selected electricians is between $25 and $30 to be approximately 0.683 or 68.3%.

Step 8 ：\(\boxed{\text{i. The probability distribution of the sample mean of hourly earnings is approximately normally distributed due to the Central Limit Theorem.}}\)

Step 9 ：\(\boxed{\text{ii. The mean and the standard deviation for the sampling distribution of the sample mean are }\mu = \$29\text{ and }\sigma = \$1.90\text{ respectively.}}\)

Step 10 ：\(\boxed{\text{iii. The probability that the average hourly earnings of the forty randomly selected electricians is between }\$25\text{ and }\$30\text{ is approximately 0.683 or 68.3%.}}\)