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

Answer 1

To solve the problem, let's first establish the parameters of the normal distribution:

- Mean (μ) = 58.1
- Standard deviation (σ) = 33.9
- Sample size (n) = 16

### 1. Find the probability that a single randomly selected value is greater than 66.6.
We need to compute \( P(X > 66.6) \). For a normal distribution, we can convert a single value to a z-score using the formula:

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

Substituting the values:

\[
z = \frac{66.6 - 58.1}{33.9} = \frac{8.5}{33.9} \approx 0.250735
\]

Now, we can look up this z-score in the standard normal distribution table or use a calculator:

\[
P(Z > 0.250735) = 1 - P(Z \leq 0.250735)
\]

From a standard normal distribution table or calculator:

\[
P(Z \leq 0.250735) \approx 0.5987
\]

Thus:

\[
P(Z > 0.250735) \approx 1 - 0.5987 = 0.4013
\]

So,

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

### 2. Find the probability that a sample of size n=16 is randomly selected with a mean greater than 66.6.
For a sample mean, the standard deviation (standard error) is given by:

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

Calculating the standard error for the sample size \( n = 16 \):

\[
\sigma_M = \frac{33.9}{\sqrt{16}} = \frac{33.9}{4} = 8.475
\]

Now, we need to calculate \( P(M > 66.6) \) using the z-score formula for the sample mean:

\[
z = \frac{M - \mu}{\sigma_M}
\]

Substituting the values:

\[
z = \frac{66.6 - 58.1}{8.475} = \frac{8.5}{8.475} \approx 1.00295
\]

Now again, we can look up this z-score:

\[
P(Z > 1.00295) = 1 - P(Z \leq 1.00295)
\]

From a standard normal distribution table or calculator:

\[
P(Z \leq 1.00295) \approx 0.8413
\]

Thus:

\[
P(Z > 1.00295) \approx 1 - 0.8413 = 0.1587
\]

So,

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

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