Let's analyze the question step-by-step using the given data:
The given table provides the statistics for sixth-grade students who are 12 years old:
- The mean height is 58 inches.
- The standard deviation of height is 2.3 inches.
### 68% Interval
For the first part, we need to determine the range where about 68% of the sixth-grade students' heights fall. This is given by the interval \((\text{mean} \pm 1 \times \text{standard deviation})\).
[tex]\[
\text{Lower bound} = 58 - 2.3 = 55.7 \text{ inches}
\][/tex]
[tex]\[
\text{Upper bound} = 58 + 2.3 = 60.3 \text{ inches}
\][/tex]
So, about 68% of sixth-grade students will have heights between 55.7 inches and 60.3 inches.
### 95% Interval
For the next part, we need the range where about 95% of the sixth-grade students' heights fall. This interval is \((\text{mean} \pm 2 \times \text{standard deviation})\).
[tex]\[
\text{Lower bound} = 58 - 2 \times 2.3 = 53.4 \text{ inches}
\][/tex]
[tex]\[
\text{Upper bound} = 58 + 2 \times 2.3 = 62.6 \text{ inches}
\][/tex]
So, about 95% of sixth-grade students will have heights between 53.4 inches and 62.6 inches.
### 99.7% Interval
Lastly, we need to determine the range where about 99.7% of the sixth-grade students' heights fall. This interval is \((\text{mean} \pm 3 \times \text{standard deviation})\).
[tex]\[
\text{Lower bound} = 58 - 3 \times 2.3 = 51.1 \text{ inches}
\][/tex]
[tex]\[
\text{Upper bound} = 58 + 3 \times 2.3 = 64.9 \text{ inches}
\][/tex]
So, about 99.7% of sixth-grade students will have heights between 51.1 inches and 64.9 inches.
### Filling in Blanks
With these calculations, we can complete the statements:
About 68% of sixth-grade students will have heights between 55.7 inches and 60.3 inches.
About 95% of sixth-grade students will have heights between 53.4 inches and 62.6 inches.
About 99.7% of sixth-grade students will have heights between 51.1 inches and 64.9 inches.
