Question

A population of values has a normal distribution with μ=58.1 and σ=33.9. You intend to draw a random sample of size n=16.
Find the probability that a single randomly selected value is greater than 66.6
P(X > 66.6)=

Find the probability that a sample of size n=16 is randomly selected with a mean greater than 66.6
P(M > 66.6) =

GIVE EXACT NUMBERS NOT ESTIMATE OR APPROX

Answer 1

To solve these problems, we will use the properties of the normal distribution.

### Part 1: Finding \( P(X > 66.6) \)

Given that the population mean (μ) is 58.1 and the population standard deviation (σ) is 33.9, we want to find the probability that a single randomly selected value \( X \) is greater than 66.6.

First, we will compute the z-score for \( X = 66.6 \):

\[
z = \frac{X - \mu}{\sigma} = \frac{66.6 - 58.1}{33.9}
\]

Calculating the numerator:

\[
66.6 - 58.1 = 8.5
\]

Now calculating the z-score:

\[
z = \frac{8.5}{33.9} \approx 0.2507
\]

Next, we look up this z-score in the standard normal distribution table, or we can use a calculator to find \( P(Z > 0.2507) \):

\[
P(Z > 0.2507) = 1 - P(Z \leq 0.2507)
\]

Using the standard normal distribution table or computational tools, we find:

\[
P(Z \leq 0.2507) \approx 0.5987
\]

Therefore,

\[
P(Z > 0.2507) \approx 1 - 0.5987 = 0.4013
\]

### Answer for Part 1:

\[
P(X > 66.6) \approx 0.4013
\]

---

### Part 2: Finding \( P(M > 66.6) \)

For a sample mean \( M \), we need to adjust the standard deviation using the formula for the standard error of the mean (SEM):

\[
SEM = \frac{\sigma}{\sqrt{n}} = \frac{33.9}{\sqrt{16}} = \frac{33.9}{4} = 8.475
\]

Now we will find the z-score for the sample mean:

\[
z = \frac{M - \mu}{SEM} = \frac{66.6 - 58.1}{8.475}
\]

Calculating the numerator:

\[
66.6 - 58.1 = 8.5
\]

Calculating the z-score:

\[
z = \frac{8.5}{8.475} \approx 1.0029
\]

Now, we look up this z-score to find \( P(Z > 1.0029) \):

\[
P(Z > 1.0029) = 1 - P(Z \leq 1.0029)
\]

Using the standard normal distribution table or computational tools, we find:

\[
P(Z \leq 1.0029) \approx 0.8413
\]

Thus,

\[
P(Z > 1.0029) \approx 1 - 0.8413 = 0.1587
\]

### Answer for Part 2:

\[
P(M > 66.6) \approx 0.1587
\]

### Summary of Results:
- \( P(X > 66.6) \approx 0.4013 \)
- \( P(M > 66.6) \approx 0.1587 \)