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Let's solve the problems step-by-step.

### Problem 1: Probability of a Single Value
Given a normal distribution with a mean \(\mu = 113.8\) and standard deviation \(\sigma = 22.9\), we need to find the probability that a single randomly selected value \(X\) falls between 109 and 113.8:

\[
P(109 < X < 113.8)
\]

**Step 1: Calculate Z-scores**

The Z-score formula is:

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

For \(X = 109\):

\[
Z_1 = \frac{109 - 113.8}{22.9} = \frac{-4.8}{22.9}
\]

For \(X = 113.8\):

\[
Z_2 = \frac{113.8 - 113.8}{22.9} = 0
\]

Let's calculate these values.

### Problem 2: Probability of a Sample Mean
Now, find the probability that a sample of size \(n = 129\) is randomly selected with a mean between 109 and 113.8:

\[
P(109 < M < 113.8)
\]

**Step 1: Calculate the Standard Error (SE)**
The standard error is given by:

\[
SE = \frac{\sigma}{\sqrt{n}} = \frac{22.9}{\sqrt{129}}
\]

**Step 2: Calculate Z-scores for the sample mean**

For \(M = 109\):

\[
Z_3 = \frac{109 - 113.8}{SE}
\]

For \(M = 113.8\):

\[
Z_4 = \frac{113.8 - 113.8}{SE} = 0
\]

I'll calculate these values now.Here are the calculations for each problem:

### Problem 1: Probability of a Single Value
- Z-score for \(X = 109\): \( Z_1 = -0.2096 \)
- Z-score for \(X = 113.8\): \( Z_2 = 0 \)
- Probability \( P(109 < X < 113.8) = 0.0830 \)

### Problem 2: Probability of a Sample Mean
- Standard Error (SE): \( SE = 2.0165 \) (calculated internally)
- Z-score for \(M = 109\): \( Z_3 = -2.3807 \)
- Z-score for \(M = 113.8\): \( Z_4 = 0 \)
- Probability \( P(109 < M < 113.8) = 0.4914 \)

Would you like further explanation on any part of the solution? 

Here are some related questions:
1. What is the importance of standard error in probability calculations?
2. How do Z-scores relate to probabilities in a normal distribution?
3. Why is the probability of a sample mean different from that of a single value?
4. How would the probabilities change if the sample size increased?
5. What would the probabilities be if the range was adjusted slightly?

**Tip:** Always double-check the Z-score calculations when dealing with normal distributions to ensure accuracy!

A population of values has a normal distribution with \( \mu = 113.8 \) and \( \sigma = 22.9 \). You intend to draw a random sample of size \( n = 129 \). Find the probability that a single randomly selected value is between 109 and 113.8. Find the probability that a sample of size \( n = 129 \) is randomly selected with a mean between 109 and 113.8.

This problem involves calculating probabilities using normal distribution and sample means. Learn how to calculate Z-scores and use the standard error to find the likelihood of specific values and sample means. Suitable for high school students studying statistics.

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