To solve this problem, we will apply the Central Limit Theorem (CLT), which allows us to approximate the sampling distribution of the sample mean. Since the sample size $n = 50$ is sufficiently large, the sampling distribution of the sample mean will be approximately normal, even if the population distribution is not.

Given information:

• Mean completion time $\mu = 38$ minutes.
• Standard deviation $\sigma = 5$ minutes.
• Sample size $n = 50$.
• We are asked to find the probability that the sample mean is more than 45 minutes.

We can calculate the probability that the sample mean is more than 45 minutes using the standard normal distribution.

Step 1: Calculate the standard error (SE) of the sample mean

The standard error of the sample mean is given by:

$SE = \frac{\sigma}{\sqrt{n}} = \frac{5}{\sqrt{50}} \approx 0.7071$

Step 2: Convert the sample mean of 45 minutes to a z-score

The z-score tells us how many standard deviations 45 minutes is from the population mean. The formula for the z-score is:

$z = \frac{\bar{x} - \mu}{SE} = \frac{45 - 38}{0.7071} = \frac{7}{0.7071} \approx 9.9$

Step 3: Find the probability

Using the z-score of 9.9, we can find the corresponding probability from the standard normal distribution table (or using a calculator). A z-score of 9.9 is extremely high, meaning the probability is virtually 0.

$P(\bar{x} > 45) \approx 0$

Conclusion:

The probability that the average completion time of the form was more than 45 minutes is practically 0.

Do you need any further details or have any questions? Here are 5 related questions for deeper understanding:

1. What is the Central Limit Theorem, and why is it important in this problem?
2. How does increasing the sample size affect the standard error of the sample mean?
3. What happens to the z-score if the standard deviation was larger?
4. Can you explain the significance of a z-score of 9.9?
5. What if the mean time for a different group was 42 minutes? How would the probability change?

Tip: When calculating probabilities involving sample means, always check if the sample size is large enough to apply the Central Limit Theorem.