Let's go through each of these probability questions using the information provided:

Given Information:

• Population mean ($\mu$) = 1200 hours
• Population standard deviation ($\sigma$) = 250 hours
• Sample size ($n$) = 100

Since we are dealing with a sample, we need to use the sampling distribution of the sample mean. For a sample size $n$, the standard deviation of the sample mean (also called the standard error) is calculated as:

$\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} = \frac{250}{\sqrt{100}} = 25$

This distribution is approximately normal because of the Central Limit Theorem (sample size is large enough).

Now, let's address each question.

Question 36: Probability that the sample mean is greater than 1150 hours

1. Calculate the z-score: $z = \frac{1150 - 1200}{25} = \frac{-50}{25} = -2$

2. Find the probability: We want $P(\bar{X} > 1150)$, which is the probability of getting a z-score greater than -2.

   Using the standard normal distribution table, $P(Z > -2)$ is equivalent to $P(Z < 2)$, which is approximately 0.9772.

   Answer for 36: 0.9772

Question 37: Probability that the sample mean is less than 1250 hours

1. Calculate the z-score: $z = \frac{1250 - 1200}{25} = \frac{50}{25} = 2$

2. Find the probability: We want $P(\bar{X} < 1250)$, which is the probability of getting a z-score less than 2.

   From the standard normal distribution, $P(Z < 2)$ is approximately 0.9772.

   Answer for 37: 0.9772

Question 38: Probability that the sample mean is between 1150 and 1250 hours

1. Calculate the z-scores for both limits:

   • For 1150: $z = \frac{1150 - 1200}{25} = -2$
   • For 1250: $z = \frac{1250 - 1200}{25} = 2$

2. Find the probability: We want $P(1150 < \bar{X} < 1250)$, which corresponds to finding the probability between $z = -2$ and $z = 2$.

   Using the standard normal distribution, $P(-2 < Z < 2) = P(Z < 2) - P(Z < -2) = 0.9772 - 0.0228 = 0.9544$.

   Answer for 38: 0.9544

Summary of Answers:

• 36. Probability greater than 1150 hours: 0.9772
• 37. Probability less than 1250 hours: 0.9772
• 38. Probability between 1150 and 1250 hours: 0.9544

Would you like a further breakdown of any steps or additional questions on this topic?

Related Questions:

1. What is the probability that the sample mean will be between 1100 and 1300 hours?
2. How does increasing the sample size to 200 change the probabilities calculated above?
3. What would the probability be if we wanted the sample mean to be greater than 1300 hours?
4. How does a decrease in standard deviation affect the probability distribution?
5. How can we interpret the standard error in the context of sample mean distribution?

Tip:

For probabilities in normally distributed data, using the z-score helps standardize any value, allowing for quick lookup in the standard normal distribution table.