To determine the range in which 95% of 7-year-old children fall in terms of height, we will use the empirical rule, also known as the 68-95-99.7 rule. This rule helps us understand that:
- Approximately 68% of the data lies within 1 standard deviation from the mean.
- Approximately 95% of the data lies within 2 standard deviations from the mean.
- Approximately 99.7% of the data lies within 3 standard deviations from the mean.
Given the data:
- Mean height (μ) = 49 inches
- Standard deviation (σ) = 2 inches
For 95% of the data, we need to calculate the range that lies within 2 standard deviations from the mean.
Step-by-step solution:
1. Calculate the lower bound:
[tex]\[
\text{Lower bound} = \text{Mean} - 2 \times \text{Standard deviation}
\][/tex]
[tex]\[
\text{Lower bound} = 49 - 2 \times 2 = 49 - 4 = 45 \text{ inches}
\][/tex]
2. Calculate the upper bound:
[tex]\[
\text{Upper bound} = \text{Mean} + 2 \times \text{Standard deviation}
\][/tex]
[tex]\[
\text{Upper bound} = 49 + 2 \times 2 = 49 + 4 = 53 \text{ inches}
\][/tex]
Therefore, based on the empirical rule, 95% of 7-year-old children are between 45 inches and 53 inches tall.
So, the completed sentence is:
"95% of 7-year-old children are between 45 inches and 53 inches tall."
