Sure! Let's solve the problem:
We are given that:
- The mean height for 7-year-old children is 49 inches.
- The standard deviation of the height is 2 inches.
According to the empirical rule (also known as the 68-95-99.7 rule):
- 68% of data lies within 1 standard deviation of the mean.
- 95% of data lies within 2 standard deviations of the mean.
- 99.7% of data lies within 3 standard deviations of the mean.
To determine the range in which 95% of the 7-year-old children fall, we need to calculate the range within 2 standard deviations of the mean.
Step 1: Calculate the lower bound of the height range.
[tex]\[ \text{Lower bound} = \text{mean} - 2 \times \text{standard deviation} \][/tex]
[tex]\[ \text{Lower bound} = 49 - 2 \times 2 \][/tex]
[tex]\[ \text{Lower bound} = 49 - 4 \][/tex]
[tex]\[ \text{Lower bound} = 45 \][/tex]
Step 2: Calculate the upper bound of the height range.
[tex]\[ \text{Upper bound} = \text{mean} + 2 \times \text{standard deviation} \][/tex]
[tex]\[ \text{Upper bound} = 49 + 2 \times 2 \][/tex]
[tex]\[ \text{Upper bound} = 49 + 4 \][/tex]
[tex]\[ \text{Upper bound} = 53 \][/tex]
Therefore, 95% of 7-year-old children are between 45 inches and 53 inches tall.
To complete the table:
[tex]\[ 95\% \text{ of 7-year-old children are between } 45 \text{ inches and } 53 \text{ inches tall.} \][/tex]
