To determine the range within which approximately 99.7% of the heights of 12-year-old sixth-grade students fall, we can use the Empirical Rule, which states that for a normal distribution:
- About 68% of the data falls within 1 standard deviation of the mean.
- About 95% of the data falls within 2 standard deviations of the mean.
- About 99.7% of the data falls within 3 standard deviations of the mean.
Given:
- Mean height (μ) = 58 inches
- Standard deviation (σ) = 2.3 inches
To find the range for 99.7% of the students' heights, we need to calculate the bounds that lie 3 standard deviations away from the mean:
1. Lower bound: [tex]\( \text{mean height} - 3 \times \text{standard deviation} \)[/tex]
2. Upper bound: [tex]\( \text{mean height} + 3 \times \text{standard deviation} \)[/tex]
Substituting the given values:
1. Lower bound = [tex]\( 58 \, \text{inches} - 3 \times 2.3 \, \text{inches} = 58 - 6.9 = 51.1 \)[/tex] inches
2. Upper bound = [tex]\( 58 \, \text{inches} + 3 \times 2.3 \, \text{inches} = 58 + 6.9 = 64.9 \)[/tex] inches
Therefore, about 99.7% of sixth-grade students will have heights between [tex]\(\square\)[/tex] inches = 51.1 inches and [tex]\(\square\)[/tex] inches = 64.9 inches.
