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


.


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


The mean x-bar=?

To find \( P_{30} \), which is the mean separating the bottom 30% of sample means from the top 70% of sample means, we need to use the properties of the sampling distribution of the sample mean. Here’s how we can proceed:

1. **Identify the given information:**
   - Population mean (\( \mu \)) = 1.197
   - Population standard deviation (\( \sigma \)) = 55.9
   - Sample size (\( n \)) = 178

2. **Determine the standard error of the mean (SEM):**
   The standard error of the mean is calculated using the formula:
   \[
   \text{SEM} = \frac{\sigma}{\sqrt{n}}
   \]
   Substituting the given values:
   \[
   \text{SEM} = \frac{55.9}{\sqrt{178}} \approx \frac{55.9}{13.34} \approx 4.19
   \]

3. **Find the z-score corresponding to the 30th percentile:**
   For a normal distribution, the z-score corresponding to the 30th percentile (P30) can be found using a z-table or standard normal distribution table. The z-score for the 30th percentile is approximately -0.524.

4. **Calculate the mean separating the bottom 30% means from the top 70% means:**
   The formula to find this mean is:
   \[
   \text{mean} = \mu + (z \times \text{SEM})
   \]
   Substituting the values:
   \[
   \text{mean} = 1.197 + (-0.524 \times 4.19) \approx 1.197 - 2.19556 \approx -0.99856
   \]

So, the mean separating the bottom 30% from the top 70% of sample means is approximately -0.999.

Would you like more details or have any questions?

Here are 5 questions you can ask next:
1. How do you calculate the z-score for other percentiles?
2. What is the relationship between population standard deviation and sample standard deviation?
3. How does the Central Limit Theorem apply to this problem?
4. Can you explain the steps to find a confidence interval for a sample mean?
5. How does increasing the sample size affect the standard error of the mean?

**Tip:** When working with normal distributions and sample means, remember that the sampling distribution of the sample mean will be normal or nearly normal if the sample size is large enough (n ≥ 30), due to the Central Limit Theorem.

Learn how to find P30 in a normal distribution using mean calculation. This problem involves calculating the mean that separates the bottom 30% of sample means from the top 70% of sample means, with specific details on population mean, standard deviation, and sample size. Suitable for undergraduate level study.

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