A population of values has a normal distribution with a population mean of 1.197 and population standard deviation of 55.9. You intend to draw a random sample of size n=178


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Find P30, which is the mean separating the bottom 30% means from the top 70% means.


The mean x-bar=?

To find \( P_{30} \) for the sample means, we need to find the value of the sample mean (\( \bar{x} \)) that separates the bottom 30% from the top 70% of the distribution of sample means.

Given:
- Population mean (\( \mu \)) = 1.197
- Population standard deviation (\( \sigma \)) = 55.9
- Sample size (\( n \)) = 178

The standard error of the mean (SEM) is given by:
\[ \text{SEM} = \frac{\sigma}{\sqrt{n}} \]

Calculating the SEM:
\[ \text{SEM} = \frac{55.9}{\sqrt{178}} \approx \frac{55.9}{13.34} \approx 4.19 \]

Next, we use the z-score corresponding to the 30th percentile. From the standard normal distribution table, the z-score for the 30th percentile is approximately -0.524.

The formula for the sample mean \( \bar{x} \) given the z-score is:
\[ \bar{x} = \mu + z \times \text{SEM} \]

Substituting the values:
\[ \bar{x} = 1.197 + (-0.524) \times 4.19 \]
\[ \bar{x} = 1.197 - 2.194 \]
\[ \bar{x} \approx -0.997 \]

Therefore, \( P_{30} \), the mean separating the bottom 30% means from the top 70% means, is approximately \(-0.997\).

Would you like more details or have any questions?

Here are some related questions you might consider:
1. How do you find the z-score for a given percentile?
2. What is the significance of the standard error of the mean?
3. How does sample size affect the standard error of the mean?
4. What is the relationship between population standard deviation and sample standard deviation?
5. How can we use normal distribution to make predictions about population parameters?

**Tip:** When working with the normal distribution, always refer to z-scores and percentiles to understand the distribution of your data relative to the mean.

Learn how to find P30, the mean that separates the bottom 30% from the top 70% of sample means in a normal distribution. This problem involves calculating the standard error of the mean, using z-scores, and understanding percentile calculations. Suitable for advanced high school or undergraduate students studying statistics.

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