To determine the range within which 95% of 7-year-old children fall according to the empirical rule, we'll consider the given mean and standard deviation of their heights.
1. Mean Height ([tex]\(\mu\)[/tex]):
- The mean height of 7-year-old children is given as 49 inches.
2. Standard Deviation ([tex]\(\sigma\)[/tex]):
- The standard deviation of their height is given as 2 inches.
3. Empirical Rule (68%-95%-99.7% Rule):
- The empirical rule states that for a normal distribution:
- Approximately 68% of the data falls within 1 standard deviation of the mean.
- Approximately 95% of the data falls within 2 standard deviations of the mean.
- Approximately 99.7% of the data falls within 3 standard deviations of the mean.
4. 95% Interval Calculation:
- To find the range where 95% of the data falls, we calculate the interval that is within 2 standard deviations from the mean.
- This means we need to calculate the lower and upper bounds as:
- [tex]\( \text{Lower Bound} = \mu - 2\sigma \)[/tex]
- [tex]\( \text{Upper Bound} = \mu + 2\sigma \)[/tex]
Let's plug in the values:
- [tex]\( \text{Lower Bound} = 49 - 2 \times 2 = 49 - 4 = 45 \)[/tex]
- [tex]\( \text{Upper Bound} = 49 + 2 \times 2 = 49 + 4 = 53 \)[/tex]
Therefore, according to the empirical rule, 95% of 7-year-old children are between 45 inches and 53 inches tall.
So the completed sentence should read:
"95% of 7-year-old children are between 45 inches and 53 inches tall."
