 Find the 60th  percentile of x . Round the answer to at least two decimal places.

To find the 60th percentile of the sample mean \(\bar{x}\) from a normally distributed population, we can use the z-score that corresponds to the 60th percentile and then convert it to the value of \(\bar{x}\).

Given:
- Population mean (\(\mu\)) = 90
- Population standard deviation (\(\sigma\)) = 24
- Sample size (n) = 86

1. Calculate the standard deviation of the sample mean \(\bar{x}\):
\[
\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} = \frac{24}{\sqrt{86}} \approx 2.5876
\]

2. Find the z-score that corresponds to the 60th percentile. For the 60th percentile, the z-score (\(z_{60}\)) is approximately 0.2533 (from the standard normal distribution table).

3. Use the z-score to find the value of \(\bar{x}\):
\[
\bar{x} = \mu + z_{60} \cdot \sigma_{\bar{x}}
\]

Let's calculate the value of \(\bar{x}\):

1. Calculate \(\sigma_{\bar{x}}\):
\[
\sigma_{\bar{x}} = \frac{24}{\sqrt{86}} \approx 2.5876
\]

2. Find the value of \(\bar{x}\) using the z-score:
\[
\bar{x} = 90 + 0.2533 \cdot 2.5876 \approx 90 + 0.6552 \approx 90.66
\]

So, the 60th percentile of \(\bar{x}\) is approximately 90.66.

The final answer, rounded to two decimal places, is:
\[
\boxed{90.66}
\]

Do you want details or have any questions?

Here are 5 additional questions you might want to ask:
1. How do you interpret percentiles in a normal distribution?
2. Can you explain how to use the z-score in finding percentiles?
3. What is the significance of the standard deviation of the sample mean?
4. How can I find different percentiles using a standard normal distribution table?
5. How do I calculate percentiles using a TI-83/84 calculator?

Tip: When finding percentiles, it’s crucial to use accurate z-scores from the standard normal distribution table to ensure precise calculations.

Learn how to find the 60th percentile of the sample mean in a normal distribution. Given population parameters and sample size, calculate the standard deviation of the sample mean, determine the z-score for the 60th percentile, and find the value of the sample mean. Suitable for advanced college-level students and professionals.

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