95% of students are between 1.1 m and 1.7 m tall. What is the mean and standard deviation?

In a normal distribution, about 95% of the data falls within two standard deviations (±2σ) of the mean (μ). This is known as the empirical rule.

### Given:
- 95% of the students are between 1.1 m and 1.7 m tall.
- Therefore, the range 1.1 m to 1.7 m represents μ - 2σ to μ + 2σ.

We can set up two equations:
- \( \mu - 2\sigma = 1.1 \)
- \( \mu + 2\sigma = 1.7 \)

### Solving for μ and σ:

1. Add both equations:  
   \[
   (\mu - 2\sigma) + (\mu + 2\sigma) = 1.1 + 1.7
   \]
   \[
   2\mu = 2.8 \quad \Rightarrow \quad \mu = 1.4 \, \text{m}
   \]

2. Subtract the first equation from the second:
   \[
   (\mu + 2\sigma) - (\mu - 2\sigma) = 1.7 - 1.1
   \]
   \[
   4\sigma = 0.6 \quad \Rightarrow \quad \sigma = 0.15 \, \text{m}
   \]

### Final answer:
- Mean (μ) = 1.4 m
- Standard deviation (σ) = 0.15 m

Would you like more details or have any questions?

Here are 5 related questions:
1. How does the empirical rule help in understanding normal distributions?
2. What percentage of students would be taller than 1.7 m?
3. How would the distribution change if the standard deviation were larger?
4. What would the range be if only 68% of the students were considered?
5. How do mean and standard deviation affect the shape of a normal distribution?

**Tip:** In a normal distribution, around 68% of data falls within 1 standard deviation of the mean, and 99.7% within 3 standard deviations.

Learn how to calculate the mean and standard deviation for a normal distribution where 95% of students are between 1.1 m and 1.7 m tall. This problem covers key concepts like the empirical rule, mean, and standard deviation, ideal for students in Grades 9-12.

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