\begin{tabular}{|c|c|c|}
\hline
Age & Mean & \begin{tabular}{c}
Standard \\
Deviation
\end{tabular} \\
\hline
7 years & 49 inches & 2 inches \\
\hline
\end{tabular}

[tex]$99.7 \%$[/tex] of 7-year-old children are between [tex]$\square$[/tex] inches and [tex]$\square$[/tex] inches tall.

According to the empirical rule:

[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.



Answer :

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.