Use the information in the table to complete the statements.

\begin{tabular}{|c|c|c|}
\hline
Age & Mean & \begin{tabular}{c}
Standard \\
Deviation
\end{tabular} \\
\hline
12 years & 58 inches & 2.3 inches \\
\hline
\end{tabular}

- About [tex]$68 \%$[/tex] of sixth-grade students will have heights between [tex]$55.7$[/tex] inches and [tex]$60.3$[/tex] inches.
- About [tex]$95 \%$[/tex] of sixth-grade students will have heights between [tex]$53.4$[/tex] inches and [tex]$62.6$[/tex] inches.
- About [tex]$99.7 \%$[/tex] of sixth-grade students will have heights between [tex]$51.1$[/tex] inches and [tex]$64.9$[/tex] inches.



Answer :

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.