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

### Given Information:
- The population has a normal distribution with a mean (\(\mu\)) of 53.9 and a standard deviation (\(\sigma\)) of 6.1.

### Part (a):
We are asked to find the probability that a single randomly selected value falls between 51.2 and 56.8. This is calculated as:

\[
P(51.2 < X < 56.8)
\]

To solve this, we need to calculate the **Z-scores** for 51.2 and 56.8 using the formula:

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

1. **For \(X = 51.2\):**

\[
Z_{51.2} = \frac{51.2 - 53.9}{6.1} = \frac{-2.7}{6.1} \approx -0.4426
\]

2. **For \(X = 56.8\):**

\[
Z_{56.8} = \frac{56.8 - 53.9}{6.1} = \frac{2.9}{6.1} \approx 0.4754
\]

Next, we find the probability corresponding to these Z-scores from the standard normal distribution table or using a calculator.

- \(P(Z = -0.4426) \approx 0.3298\)
- \(P(Z = 0.4754) \approx 0.6823\)

Finally, the probability is:

\[
P(51.2 < X < 56.8) = P(Z = 0.4754) - P(Z = -0.4426) \approx 0.6823 - 0.3298 = 0.3525
\]

So, the probability for part (a) is approximately **0.3525**.

### Part (b):
For part (b), we are asked to find the probability that a randomly selected sample of size \(n = 44\) has a mean between 51.2 and 56.8. This requires the **standard error** of the mean, which is calculated as:

\[
\text{Standard Error} = \frac{\sigma}{\sqrt{n}} = \frac{6.1}{\sqrt{44}} \approx \frac{6.1}{6.6332} \approx 0.9194
\]

Now, we calculate the Z-scores for the sample mean using this standard error:

1. **For \(M = 51.2\):**

\[
Z_{51.2} = \frac{51.2 - 53.9}{0.9194} = \frac{-2.7}{0.9194} \approx -2.9362
\]

2. **For \(M = 56.8\):**

\[
Z_{56.8} = \frac{56.8 - 53.9}{0.9194} = \frac{2.9}{0.9194} \approx 3.1547
\]

We now find the probabilities corresponding to these Z-scores:

- \(P(Z = -2.9362) \approx 0.0017\)
- \(P(Z = 3.1547) \approx 0.9992\)

So, the probability is:

\[
P(51.2 < M < 56.8) = P(Z = 3.1547) - P(Z = -2.9362) \approx 0.9992 - 0.0017 = 0.9975
\]

Thus, the probability for part (b) is approximately **0.9975**.

---

Let me know if you'd like more details or have any questions!

### Related questions:
1. What are Z-scores, and how are they used in probability calculations?
2. How does sample size affect the standard error in normal distributions?
3. How can we calculate probabilities for non-standard normal distributions?
4. What is the central limit theorem and how does it relate to sample means?
5. How do we interpret probabilities in the context of normal distributions?

### Tip:
Always check if you're working with a population or sample, as this affects whether you need to use the standard error or population standard deviation when calculating probabilities!

A population of values has a normal distribution with μ = 53.9 and σ = 6.1. a) Find the probability that a single randomly selected value is between 51.2 and 56.8. b) Find the probability that a randomly selected sample of size n = 44 has a mean between 51.2 and 56.8.

Find the probability that a single value or sample mean from a normal distribution falls between specific points. This problem involves calculating Z-scores and using the standard error for a sample size of n = 44. The population mean is 53.9, and the standard deviation is 6.1.

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