To solve this problem using the empirical rule, let's follow these steps:
1. Understand the Empirical Rule: The empirical rule states that for a normal distribution:
- Approximately 68% of the data falls within one standard deviation (σ) of the mean (μ).
- Approximately 95% falls within two standard deviations.
- Approximately 99.7% falls within three standard deviations.
2. Given Data:
- Mean height (μ) for 7-year-old children: 49 inches
- Standard deviation (σ): 2 inches
3. Apply the Empirical Rule for 68%:
- To find the range that includes 68% of the data, calculate one standard deviation above and below the mean.
- Lower bound: Mean height (μ) - Standard deviation (σ)
- Upper bound: Mean height (μ) + Standard deviation (σ)
4. Compute the Bounds:
- Lower bound: [tex]\( 49 \, \text{inches} - 2 \, \text{inches} = 47 \, \text{inches} \)[/tex]
- Upper bound: [tex]\( 49 \, \text{inches} + 2 \, \text{inches} = 51 \, \text{inches} \)[/tex]
Putting it all together, 68% of 7-year-old children are between 47 inches and 51 inches tall.
