Question

A population of values has a normal distribution with μ=65.4 and o=61.9. You intend to draw a random sample of size n=24.

Find the probability that a single randomly selected value is less than 97.
P(X < 97) =


Find the probability that a sample of size n=24 is randomly selected with a mean less than 97.
P(M < 97) =


Enter your answers as numbers accurate to 4 decimal places. Answers obtained using exact z-scores or z-scores rounded to 3 decimal places are accepted.

Answer 1

To find the required probabilities, we will use the properties of the normal distribution and the z-score formula.

1. **Finding the probability that a single randomly selected value is less than 97 (P(X < 97)):**

Given:
- Mean (μ) = 65.4
- Standard deviation (σ) = 61.9
- Value (X) = 97

We first calculate the z-score, which is given by the formula:

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

Substituting the values:

\[
z = \frac{97 - 65.4}{61.9} = \frac{31.6}{61.9} \approx 0.5097
\]

Now, we will look up the z-score in the standard normal distribution table or use a calculator to find the probability associated with this z-score.

Using a z-table or calculator:

\[
P(X < 97) \approx P(Z < 0.5097) \approx 0.6943
\]

2. **Finding the probability that a sample of size n=24 has a mean less than 97 (P(M < 97)):**

When dealing with sample means, we need to consider the standard error (SE), which is calculated as:

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

Where:
- n = 24
- σ = 61.9

Calculating the standard error:

\[
SE = \frac{61.9}{\sqrt{24}} \approx \frac{61.9}{4.899} \approx 12.629
\]

Next, we calculate the z-score for the sample mean using the same formula as before:

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

Substituting the values:

\[
z = \frac{97 - 65.4}{12.629} = \frac{31.6}{12.629} \approx 2.5043
\]

Now, we find the probability associated with this z-score:

\[
P(M < 97) \approx P(Z < 2.5043)
\]

Using a z-table or calculator:

\[
P(Z < 2.5043) \approx 0.9939
\]

3. **Final Answers:**

Therefore,
- \( P(X < 97) \approx 0.6943 \)
- \( P(M < 97) \approx 0.9939 \)

Final answers (accurate to 4 decimal places):
- **P(X < 97) = 0.6943**
- **P(M < 97) = 0.9939**