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*53, which is the value (*X*) separating the bottom 53% values from the top 47% values.
*P*53 (for population) =

Find *P*53, which is the sample mean () separating the bottom 53% sample means from the top 47% sample means.
*P*53 (for sample means) =

We are given a normally distributed population with:

- Population mean \( \mu \)
- Population standard deviation \( \sigma \)
- Sample size \( n \)

### Step 1: Find \( P_{53} \) for the population

We need to find the value \( P_{53} \), which separates the bottom 53% of the population from the top 47%.

For a standard normal distribution, the z-score corresponding to a cumulative probability of 0.53 can be found using a standard normal table or a calculator.

#### Z-Score Calculation:
Using a z-table or a standard normal calculator, the z-score corresponding to 0.53 is approximately:

\[
z = 0.075
\]

The formula to find the value of \( X \) (population) is:

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

Substitute \( z = 0.075 \), \( \mu = \mu \), and \( \sigma = \sigma \):

\[
P_{53} \text{ for the population} = \mu + 0.075 \cdot \sigma
\]

### Step 2: Find \( P_{53} \) for the sample means

To find \( P_{53} \) for the sample means, we need to calculate the standard error of the mean, which is given by:

\[
SE = \frac{\sigma}{\sqrt{n}}
\]

Next, use the same z-score (0.075) to calculate \( P_{53} \) for the sample means. The formula for the sample mean is:

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

Substitute \( SE = \frac{\sigma}{\sqrt{n}} \):

\[
P_{53} \text{ for the sample means} = \mu + 0.075 \cdot \frac{\sigma}{\sqrt{n}}
\]

Now that we have the formulas, let me know if you'd like to input specific values for \( \mu \), \( \sigma \), and \( n \), so we can proceed with the calculations and provide the final values to 1 decimal place.

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 P53, which is the value (X) separating the bottom 53% values from the top 47% values. P53 (for population) = Find P53, which is the sample mean separating the bottom 53% sample means from the top 47% sample means. P53 (for sample means) =

Solve for P53, the value separating the bottom 53% of values in a normal distribution from the top 47%, for both population and sample means. Learn the key statistical concepts of z-scores, standard deviation, and the Central Limit Theorem. Suitable for Grades 11-12 or introductory college statistics.

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