Question

A population of values has a normal distribution with μ=21 and σ=33.7. You intend to draw a random sample of size n=15.
Find the probability that a single randomly selected value is greater than 33.2
P(X > 33.2)=

Find the probability that a sample of size n=15 is randomly selected with a mean greater than 33.2
P(M > 33.2) =

Enter your answers as numbers accurate

Answer 1

To find the probabilities requested, we'll use the properties of the normal distribution.

### Part 1: Probability that a single randomly selected value is greater than 33.2

The given population parameters are:
- Mean (μ) = 21
- Standard deviation (σ) = 33.7

We want to find:
\[ P(X > 33.2) \]

First, we need to calculate the Z-score for \( x = 33.2 \):

\[
Z = \frac{X - \mu}{\sigma} = \frac{33.2 - 21}{33.7} = \frac{12.2}{33.7} \approx 0.3629
\]

Now, we can look up the Z-score in the standard normal distribution table or use a calculator to find \( P(Z < 0.3629) \).

Using a Z-table or calculator, we find:

\[
P(Z < 0.3629) \approx 0.6436
\]

Thus, to find \( P(X > 33.2) \):

\[
P(X > 33.2) = 1 - P(Z < 0.3629) \approx 1 - 0.6436 = 0.3564
\]

### Part 2: Probability that a sample of size n=15 has a mean greater than 33.2

For the sample mean, we will use the Central Limit Theorem. The mean of the sample means (\( \mu_M \)) remains \( \mu = 21 \), but the standard deviation of the sample means (standard error, \( \sigma_M \)) is calculated as:

\[
\sigma_M = \frac{\sigma}{\sqrt{n}} = \frac{33.7}{\sqrt{15}} \approx \frac{33.7}{3.8729} \approx 8.693
\]

Now, we want to find:
\[ P(M > 33.2) \]

We calculate the Z-score for the sample mean:

\[
Z = \frac{M - \mu_M}{\sigma_M} = \frac{33.2 - 21}{8.693} = \frac{12.2}{8.693} \approx 1.402
\]

Now we can look up the Z-score in the standard normal distribution table or use a calculator to find \( P(Z < 1.402) \).

Using a Z-table or calculator, we find:

\[
P(Z < 1.402) \approx 0.9191
\]

Thus, to find \( P(M > 33.2) \):

\[
P(M > 33.2) = 1 - P(Z < 1.402) \approx 1 - 0.9191 = 0.0809
\]

### Final Answers:
- \( P(X > 33.2) \approx 0.3564 \)
- \( P(M > 33.2) \approx 0.0809 \)

Therefore:
- \( P(X > 33.2) = 0.3564 \)
- \( P(M > 33.2) = 0.0809 \)