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We are dealing with a normal distribution for the heights of 1000 students, where:

- Mean (μ) = 174.5 cm
- Standard deviation (σ) = 6.9 cm
- Sample size = 25 (n)
- 200 random samples are drawn.

The problem is asking us to determine:

a) The number of sample means that fall between 172.5 cm and 175.8 cm.

b) The number of sample means that fall below 172 cm.

### To solve this, we need to follow these steps:

1. **Calculate the standard error (SE)** of the mean using the formula:
   \[
   SE = \frac{\sigma}{\sqrt{n}} = \frac{6.9}{\sqrt{25}} = 6.9 / 5 = 1.38
   \]

2. **Convert the values to Z-scores** using the Z-score formula:
   \[
   Z = \frac{X - \mu}{SE}
   \]
   where \(X\) is the value we are interested in, \(\mu\) is the population mean, and \(SE\) is the standard error.

3. **Find the probability corresponding to the Z-scores** using the standard normal distribution table.

4. **Calculate the number of samples** by multiplying the probability by the total number of samples (200).

---

### Part a) Sample means between 172.5 cm and 175.8 cm:

- First, calculate the Z-scores for 172.5 cm and 175.8 cm.

For 172.5 cm:
\[
Z_{172.5} = \frac{172.5 - 174.5}{1.38} = \frac{-2}{1.38} \approx -1.45
\]

For 175.8 cm:
\[
Z_{175.8} = \frac{175.8 - 174.5}{1.38} = \frac{1.3}{1.38} \approx 0.94
\]

Now, using the Z-table, we find the probabilities:
- \(P(Z = -1.45) \approx 0.0735\)
- \(P(Z = 0.94) \approx 0.8264\)

The probability between these Z-scores is:
\[
P(172.5 < X < 175.8) = P(0.8264) - P(0.0735) = 0.8264 - 0.0735 = 0.7529
\]

Thus, the number of samples with means between 172.5 cm and 175.8 cm is:
\[
0.7529 \times 200 \approx 150.58 \approx 151 \text{ samples}
\]

---

### Part b) Sample means below 172 cm:

For 172 cm:
\[
Z_{172} = \frac{172 - 174.5}{1.38} = \frac{-2.5}{1.38} \approx -1.81
\]

Using the Z-table:
\[
P(Z = -1.81) \approx 0.0351
\]

Thus, the number of samples with means below 172 cm is:
\[
0.0351 \times 200 \approx 7.02 \approx 7 \text{ samples}
\]

---

### Final Answer:

a) Approximately **151 sample means** fall between 172.5 cm and 175.8 cm.

b) Approximately **7 sample means** fall below 172 cm.

Would you like further clarification or more details?

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Here are 5 related questions to deepen your understanding:

1. How does increasing the sample size affect the standard error of the mean?
2. What is the significance of Z-scores in a normal distribution?
3. How would the results change if the population standard deviation was larger?
4. What is the central limit theorem, and how does it apply to this problem?
5. How could we use confidence intervals to estimate the range of sample means?

**Tip**: When solving problems with normal distributions, always start by standardizing the values (using Z-scores) to make use of the standard normal distribution.

Determine the number of sample means that fall between 172.5 and 175.8 cm and the number of sample means that fall below 172 cm, given a population mean of 174.5 cm, standard deviation of 6.9 cm, and sample size of 25 with 200 samples.

This detailed solution covers how to calculate the number of sample means that fall between 172.5 cm and 175.8 cm, and the number below 172 cm. Using the concepts of Z-scores and standard error with a normal distribution, this problem is ideal for students in advanced statistics courses.

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