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This problem provides information about a normally distributed population of women’s heights and asks for an estimate of the mean and standard deviation using the empirical rule (also known as the 68-95-99.7 rule). Let's break it down step by step.

### Step 1: Interpreting the percentages with respect to the empirical rule
- **16% of women have heights less than 62 inches.**
- **97.5% of women have heights less than 71 inches.**

The empirical rule tells us:
- 68% of the data falls within 1 standard deviation (σ) from the mean (μ),
- 95% of the data falls within 2 standard deviations from the mean,
- 99.7% of the data falls within 3 standard deviations from the mean.

### Step 2: Using the Z-scores for the given percentages
We can match the given percentages to corresponding Z-scores:
- A **16%** percentile corresponds roughly to a **Z-score of -1** (1 standard deviation below the mean).
- A **97.5%** percentile corresponds to a **Z-score of +2** (2 standard deviations above the mean).

### Step 3: Setting up the equations
We know:
1. 62 inches corresponds to 1 standard deviation below the mean, i.e., \( 62 = \mu - \sigma \).
2. 71 inches corresponds to 2 standard deviations above the mean, i.e., \( 71 = \mu + 2\sigma \).

### Step 4: Solving the system of equations
We now solve the two equations:
1. \( 62 = \mu - \sigma \)
2. \( 71 = \mu + 2\sigma \)

Let's solve this system to find the mean (μ) and standard deviation (σ).The solution to the system of equations gives:

- The **mean height (μ)** is **65 inches**.
- The **standard deviation (σ)** is **3 inches**.

Would you like further details or have any questions?

Here are 5 related questions you might find interesting:
1. How would the results change if a different percentile were given?
2. Can the empirical rule be used for non-normal distributions?
3. How can Z-scores be calculated for any given percentage?
4. What would be the range of heights that include 95% of the population?
5. How does changing the standard deviation affect the spread of the data?

**Tip:** The empirical rule is a great shortcut for estimating values, but it only applies to data that is normally distributed.

The distribution of heights in a population of women is approximately normal. Sixteen percent of the women have heights less than 62 inches. About 97.5% of the women have heights less than 71 inches. Use the empirical rule to estimate the mean and standard deviation of the heights in this population.

Learn how to estimate the mean and standard deviation of heights in a population of women based on given percentiles using the empirical rule. The problem covers normal distribution, Z-scores, and a step-by-step solution for solving the system of equations.

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