According to the empirical rule for a normal distribution, $99.7\%$ of $7$-year-old children are between which of the following heights if the mean height is $48$ inches and the standard deviation is $2$ inches?

• A. Between $40$ inches and $56$ inches
• B. Between $46$ inches and $50$ inches
• C. Between $42$ inches and $54$ inches
• D. Between $44$ inches and $52$ inches

Answer: (C)

Verified step by step guidance

1. Recall the empirical rule (68-95-99.7 rule) for a normal distribution, which states that approximately 99.7% of the data lies within three standard deviations from the mean.
2. Identify the mean (\(\mu\)) and standard deviation (\(\sigma\)) given in the problem: \(\mu = 48\) inches and \(\sigma = 2\) inches.
3. Calculate the lower bound of the height range by subtracting three times the standard deviation from the mean: \(\mu - 3\sigma = 48 - 3 \times 2\).
4. Calculate the upper bound of the height range by adding three times the standard deviation to the mean: \(\mu + 3\sigma = 48 + 3 \times 2\).
5. Interpret the results to find the interval that contains approximately 99.7% of the heights, which corresponds to the range between the lower and upper bounds calculated.