Question

A population of values has a normal distribution with μ=21 and o=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 to 4 decimal places.

Answer 1

To calculate the probabilities you've requested, we need to use the properties of the normal distribution.

1. **Finding P(X > 33.2):**

We are looking for the probability that a single randomly selected value (X) is greater than 33.2. Given that the population mean (μ) is 21 and the standard deviation (σ) is 33.7, we first compute the z-score for X = 33.2.

The formula for the z-score is:
\[
z = \frac{X - \mu}{\sigma}
\]

Plugging in the values:
\[
z = \frac{33.2 - 21}{33.7} = \frac{12.2}{33.7} \approx 0.362
\]

Next, we look up the z-score in the standard normal distribution table or use a calculator to find P(Z > 0.362).

The cumulative probability for Z = 0.362 is approximately 0.6438, so:
\[
P(X > 33.2) = 1 - P(Z \leq 0.362) = 1 - 0.6438 = 0.3562
\]

2. **Finding P(M > 33.2) for a sample of size n = 15:**

When sampling from a population, the distribution of the sample mean (M) will also be normally distributed with mean μ and standard deviation (standard error) calculated as follows:
\[
\sigma_M = \frac{\sigma}{\sqrt{n}} = \frac{33.7}{\sqrt{15}} \approx \frac{33.7}{3.872} \approx 8.692
\]

Now we again calculate the z-score for M = 33.2:
\[
z = \frac{M - \mu}{\sigma_M} = \frac{33.2 - 21}{8.692} = \frac{12.2}{8.692} \approx 1.402
\]

We now find the probability:
\[
P(M > 33.2) = 1 - P(Z \leq 1.402)
\]

Checking the z-table for P(Z ≤ 1.402), we find it approximately equals 0.9192. Therefore:
\[
P(M > 33.2) = 1 - 0.9192 = 0.0808
\]

So, rounding to four decimal places, our results are:

1. \( P(X > 33.2) \approx 0.3562 \)
2. \( P(M > 33.2) \approx 0.0808 \)

Final answers:

- \( P(X > 33.2) = 0.3562 \)
- \( P(M > 33.2) = 0.0808 \)