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Answer :
Sure! Let's solve this step-by-step.
1. Understanding the Problem:
We are told that 99.7% of American men's heights fall between 5 feet 0 inches (5′0″) and 7 feet 0 inches (7′0″). We need to estimate the standard deviation of the height distribution.
2. Converting Heights to Inches:
To make calculations easier, we'll convert the heights from feet and inches to total inches.
- 5 feet 0 inches = 5 12 + 0 = 60 inches
- 7 feet 0 inches = 7 12 + 0 = 84 inches
3. Using the Empirical Rule:
According to the empirical rule (68-95-99.7 rule), approximately 99.7% of the data in a normal distribution lies within three standard deviations (3σ) on either side of the mean (μ).
Therefore, the interval from 5′0″ to 7′0″ (which is 60 inches to 84 inches) represents a span of 6σ (since it covers three standard deviations on each side of the mean).
4. Calculating the Total Range:
The total range of heights given is:
[tex]\[
\text{Range} = 84 \text{ inches} - 60 \text{ inches} = 24 \text{ inches}
\][/tex]
5. Finding the Standard Deviation:
Since this 24-inch range represents 6σ, we can find σ by dividing the range by 6:
[tex]\[
\sigma = \frac{24 \text{ inches}}{6} = 4 \text{ inches}
\][/tex]
So, the standard deviation of the height of American men is 4 inches.
The correct answer is:
[tex]\[
\boxed{4^{\prime \prime}}
\][/tex]
1. Understanding the Problem:
We are told that 99.7% of American men's heights fall between 5 feet 0 inches (5′0″) and 7 feet 0 inches (7′0″). We need to estimate the standard deviation of the height distribution.
2. Converting Heights to Inches:
To make calculations easier, we'll convert the heights from feet and inches to total inches.
- 5 feet 0 inches = 5 12 + 0 = 60 inches
- 7 feet 0 inches = 7 12 + 0 = 84 inches
3. Using the Empirical Rule:
According to the empirical rule (68-95-99.7 rule), approximately 99.7% of the data in a normal distribution lies within three standard deviations (3σ) on either side of the mean (μ).
Therefore, the interval from 5′0″ to 7′0″ (which is 60 inches to 84 inches) represents a span of 6σ (since it covers three standard deviations on each side of the mean).
4. Calculating the Total Range:
The total range of heights given is:
[tex]\[
\text{Range} = 84 \text{ inches} - 60 \text{ inches} = 24 \text{ inches}
\][/tex]
5. Finding the Standard Deviation:
Since this 24-inch range represents 6σ, we can find σ by dividing the range by 6:
[tex]\[
\sigma = \frac{24 \text{ inches}}{6} = 4 \text{ inches}
\][/tex]
So, the standard deviation of the height of American men is 4 inches.
The correct answer is:
[tex]\[
\boxed{4^{\prime \prime}}
\][/tex]
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