A population of values has a normal distribution with population standard deviation of 151.6 and standard deviation of 72.5 You intend to draw a random sample of size n=121.

Find P30, which is the mean separating the bottom 32% means from the top 68% means.

To find \( P_{30} \) for the sample means, which separates the bottom 30% of the sample means from the top 70%, we can use the properties of the normal distribution and the concept of the sampling distribution of the sample mean.

Given dataTo find \( P_{30} \) for the sample means, which separates the bottom 30% of the sample means from the top 70%, we can use the properties of the normal distribution and the concept of the sampling distribution of the sample mean.

Given 
- Population standard deviation, \(\sigma = 151.6\)
- Sample size, \(n = 121\)

The standard deviation of the sampling distribution of the sample mean (standard error) is given by:
\[ \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} \]

Substituting the given values:
\[ \sigma_{\bar{x}} = \frac{151.6}{\sqrt{121}} = \frac{151.6}{11} \approx 13.78 \]

Next, we need to find the z-score that corresponds to the bottom 30% of the distribution. This z-score can be found using standard normal distribution tables or a z-score calculator. The z-score that separates the bottom 30% from the top 70% is approximately:
\[ z \approx -0.524 \]

Now, using the z-score formula:
\[ z = \frac{X - \mu_{\bar{x}}}{\sigma_{\bar{x}}} \]

We can solve for \( X \) (which represents \( P_{30} \)) as follows:
\[ -0.524 = \frac{P_{30} - \mu_{\bar{x}}}{13.78} \]

To find \( P_{30} \), we need the population mean (\(\mu_{\bar{x}}\)). Since it’s not given, we assume \(\mu_{\bar{x}} = \mu \), the population mean. However, we can still solve for \( P_{30} \) relative to the population mean:
\[ P_{30} = \mu + (-0.524 \times 13.78) \]
\[ P_{30} = \mu - 7.22 \]

Therefore, the value of \( P_{30} \) that separates the bottom 30% of the sample means from the top 70% of the sample means is \(\mu - 7.22\), where \(\mu\) is the population mean.

If you have any specific value for the population mean \(\mu\), you can plug it into the equation to find the exact \( P_{30} \).

Would you like to know more details or have any questions about this problem? Here are some related questions you might find helpful:

1. How do you interpret the z-score in a normal distribution?
2. What are the properties of the sampling distribution of the sample mean?
3. How do you find percentiles in a normal distribution?
4. What is the significance of the standard error in statistics?
5. How does sample size affect the sampling distribution of the sample mean?

**Tip:** When dealing with the normal distribution, understanding the z-score and how it relates to percentiles is crucial for finding probabilities and critical values.

Learn how to find P30 in a normal distribution and sampling distribution problem. Calculate the value separating the bottom 30% of sample means from the top 70%. Includes step-by-step solution using the standard error of the mean (SEM) and z-score. Suitable for advanced college-level statistics.

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