A population of values has a normal distribution with  and . You intend to draw a random sample of size . Please answer the following questions, and **show your answers to 1 decimal place**.

Find *P*18, which is the value (*X*) separating the bottom 18% values from the top 82% values. 
*P*18 (for population) = 

Find *P*18, which is the sample mean () separating the bottom 18% sample means from the top 82% sample means. 
*P*18 (for sample means) = 




We are given a population with a normal distribution where the population mean is \(\mu\) and the population standard deviation is \(\sigma\). The sample size is \(n\). We are tasked with finding the value that separates the bottom 18% from the top 82% of values in both the population and the sample means.

### Given Information:
- Population distribution: Normal with \(\mu\) (mean) and \(\sigma\) (standard deviation).
- Sample size: \(n\).

### Step 1: **Finding \( P_{18} \) for the population**

To find \( P_{18} \), which is the value separating the bottom 18% from the top 82%, we need to use the standard normal distribution (z-distribution). The corresponding z-score for the bottom 18% can be found using the z-table or a standard normal distribution calculator.

1. Look for the z-score that corresponds to a cumulative probability of 0.18 (18%) in the z-table.

   The z-score for 0.18 is approximately **-0.915**.

2. Now, convert the z-score back to the original value \( P_{18} \) using the formula for a normal distribution:

   \[
   X = \mu + z \cdot \sigma
   \]

   where:
   - \(\mu\) is the population mean,
   - \(\sigma\) is the population standard deviation,
   - \(z\) is the z-score for 18%.

### Step 2: **Finding \( P_{18} \) for the sample means**

The distribution of the sample mean \(\bar{X}\) is also normally distributed, with mean \(\mu\) and standard deviation \(\sigma_{\bar{X}}\) (the standard error of the mean). The standard error is calculated as:

\[
\sigma_{\bar{X}} = \frac{\sigma}{\sqrt{n}}
\]

We will use the same z-score of **-0.915** since we are still looking for the value separating the bottom 18% from the top 82%, but this time, the formula will be:

\[
\bar{X} = \mu + z \cdot \sigma_{\bar{X}}
\]

Now, let’s compute the actual values once we have the population mean, standard deviation, and sample size \(n\).

Please provide the values of \(\mu\), \(\sigma\), and \(n\) so we can proceed with the calculations.

A population of values has a normal distribution with μ and σ. You intend to draw a random sample of size n. Please answer the following questions, and show your answers to 1 decimal place: Find P18, which is the value (X) separating the bottom 18% values from the top 82% values. P18 (for population) = Find P18, which is the sample mean (x̄) separating the bottom 18% sample means from the top 82% sample means. P18 (for sample means) =

Learn how to calculate P18 for both the population and sample means in a normally distributed population. This problem involves using z-scores, the normal distribution, and the central limit theorem to separate the bottom 18% from the top 82% values. Suitable for college-level statistics students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation