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Use the information in the table to complete the statements.

| Age | Mean | Standard Deviation |
|----------|----------|--------------------|
| 12 years | 58 inches | 2.3 inches |

1. About 68% of sixth-grade students will have heights between 55.7 inches and 60.3 inches.
2. About 95% of sixth-grade students will have heights between 53.4 inches and 62.6 inches.
3. About 99.7% of sixth-grade students will have heights between ____ inches and ____ inches.


Sagot :

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.