To solve this problem, we will apply the Empirical Rule (68-95-99.7 Rule) for a normal distribution. Given the parameters:
- Mean height ([tex]\(\mu\)[/tex]) = 49 inches
- Standard deviation ([tex]\(\sigma\)[/tex]) = 2 inches
Step-by-step solution for the ranges corresponding to the Empirical Rule:
1. For 68% of the data:
- This range includes data within 1 standard deviation from the mean.
- Lower bound: [tex]\( \mu - \sigma = 49 - 2 = 47 \)[/tex] inches
- Upper bound: [tex]\( \mu + \sigma = 49 + 2 = 51 \)[/tex] inches
So, 68% of 7-year-old children are between 47 inches and 51 inches tall.
2. For 95% of the data:
- This range covers data within 2 standard deviations from the mean.
- Lower bound: [tex]\( \mu - 2\sigma = 49 - 2 \times 2 = 45 \)[/tex] inches
- Upper bound: [tex]\( \mu + 2\sigma = 49 + 2 \times 2 = 53 \)[/tex] inches
So, 95% of 7-year-old children are between 45 inches and 53 inches tall.
3. For 99.7% of the data:
- This range contains data within 3 standard deviations from the mean.
- Lower bound: [tex]\( \mu - 3\sigma = 49 - 3 \times 2 = 43 \)[/tex] inches
- Upper bound: [tex]\( \mu + 3\sigma = 49 + 3 \times 2 = 55 \)[/tex] inches
So, 99.7% of 7-year-old children are between 43 inches and 55 inches tall.
Summary of the ranges:
1. 68% of 7-year-old children are between 47 inches and 51 inches tall.
2. 95% of 7-year-old children are between 45 inches and 53 inches tall.
3. 99.7% of 7-year-old children are between 43 inches and 55 inches tall.
Thus:
- [tex]$99.7 \%$[/tex] of 7-year-old children are between 43 inches and 55 inches tall.
- [tex]$68 \%$[/tex] of 7-year-old children are between 47 inches and 51 inches tall.
- [tex]$95 \%$[/tex] of 7-year-old children are between 45 inches and 53 inches tall.
