This problem involves the sampling distribution of the sample mean. We need to compute the probability that the sample mean completion time is more than 45 minutes.

Since the population mean $\mu = 38$ minutes and the population standard deviation $\sigma = 5$ minutes are given, we can use the central limit theorem (CLT). For a sufficiently large sample (in this case $n = 50$), the sampling distribution of the sample mean follows a normal distribution with the following characteristics:

• Mean of the sampling distribution: $\mu_{\bar{X}} = \mu = 38$
• Standard error (SE) of the mean:
  $\sigma_{\bar{X}} = \frac{\sigma}{\sqrt{n}} = \frac{5}{\sqrt{50}} = \frac{5}{7.071} \approx 0.707$

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Step 1: Convert 45 minutes to a z-score

We want to calculate the probability that the sample mean is more than 45 minutes. For that, we convert the value of 45 minutes into a z-score:

$z = \frac{X - \mu_{\bar{X}}}{\sigma_{\bar{X}}} = \frac{45 - 38}{0.707} = \frac{7}{0.707} \approx 9.9$

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Step 2: Find the probability using the z-table

A z-score of 9.9 is extremely high. The area to the left of this z-score (which corresponds to the cumulative probability) is effectively 1. That means the probability that the sample mean is greater than 45 minutes is:

$P(\bar{X} > 45) = 1 - P(Z < 9.9) \approx 1 - 1 = 0$

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Final Answer

The probability that the average completion time for the sample of 50 people is more than 45 minutes is approximately:

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

This result makes sense since 45 minutes is far above the population mean (38 minutes), and with a small standard error of 0.707, it's extremely unlikely that a random sample of 50 people would take that long on average.

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Would you like a deeper explanation or have any follow-up questions?
Here are some related questions you might find helpful:

1. What is the central limit theorem and why is it important?
2. How do you calculate the standard error for different sample sizes?
3. What is the purpose of a z-score in hypothesis testing?
4. How can we use a normal distribution table for probability calculations?
5. How does increasing the sample size affect the standard error and the probability?

Tip: In practice, probabilities that yield extremely high z-scores (like 9.9) are treated as 0, because they are beyond the range of meaningful outcomes.