Let's break down each part of the problem step by step.

Part 1: Compute the Standard Error (SE)

We are given the formula for the Standard Error (SE) of the mean:

$\text{SE} = \frac{\sigma}{\sqrt{n}}$

Where:

• $\sigma = 60.4$ (population standard deviation)
• $n = 226$ (sample size)

So, the standard error is:

$\text{SE} = \frac{60.4}{\sqrt{226}} = \frac{60.4}{15.033} \approx 4.02$

Thus, the Standard Error (SE) is approximately 4.02.

Part 2: Compute the z-score

The formula for the z-score is:

$z = \frac{\bar{X} - \mu}{\text{SE}}$

Where:

• $\bar{X} = 53.7$ (sample mean)
• $\mu = 63.7$ (population mean)

Now, plug the values into the z-score formula:

$z = \frac{53.7 - 63.7}{4.02} = \frac{-10}{4.02} \approx -2.49$

Thus, the z-score is approximately -2.49.

Part 3: Find the probability $P(Z \leq z)$

To find the probability associated with a z-score of -2.49, we can use a standard normal table or a calculator.

Looking up $z = -2.49$ in the standard normal table or using a calculator, we find:

$P(Z \leq -2.49) \approx 0.0064$

Thus, the probability that $Z \leq -2.49$ is approximately 0.0064.

Part 4: Compute $P(\bar{X} > 53.7)$

Now, we can compute the probability that $\bar{X} > 53.7$ using the relation:

$P(\bar{X} > 53.7) = 1 - P(Z \leq -2.49)$

So:

$P(\bar{X} > 53.7) = 1 - 0.0064 = 0.9936$

Thus, the probability that $\bar{X} > 53.7$ is approximately 0.9936.

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Final Answers:

1. Standard Error (SE): 4.02
2. z-score: -2.49
3. P(Z ≤ z): 0.0064
4. P(X̄ > 53.7): 0.9936

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Do you have any questions about the steps or need further clarification?

Here are 5 related questions you might find interesting:

1. How does the standard error affect the precision of a sample estimate?
2. What is the relationship between the z-score and probability in hypothesis testing?
3. How do you interpret negative z-scores in context?
4. How can you use the standard normal distribution to calculate confidence intervals?
5. How does sample size $n$ affect the standard error?

Tip: The z-score is a great tool for understanding how far a data point is from the mean, measured in terms of the standard deviation! The further the z-score, the more extreme the value.