Question

3.2 measures of spread: empirical rule
the army finds that the head sizes (forehead circumference) of soldiers has a normal distribution with a mean of 22.6 inches and a standard deviation of 0.9 inches.

1. approximately 68% of soldiers have head sizes between ( ) inches.
2. approximately 95% of soldiers have head sizes between ( ) inches.
3. approximately 99.7% of soldiers have head sizes between ( ) inches.

Explanation:

Step1: Recall empirical - rule for normal distribution

The empirical rule states that for a normal distribution:

• Approximately 68% of the data lies within 1 standard - deviation of the mean, i.e., $\mu\pm\sigma$.
• Approximately 95% of the data lies within 2 standard - deviations of the mean, i.e., $\mu\pm2\sigma$.
• Approximately 99.7% of the data lies within 3 standard - deviations of the mean, i.e., $\mu\pm3\sigma$.

Given $\mu = 22.6$ inches and $\sigma=0.9$ inches.

Step2: Calculate for 68% range

For approximately 68% of the data:
Lower limit: $\mu-\sigma=22.6 - 0.9=21.7$ inches
Upper limit: $\mu+\sigma=22.6 + 0.9=23.5$ inches

Step3: Calculate for 95% range

For approximately 95% of the data:
Lower limit: $\mu - 2\sigma=22.6-2\times0.9=22.6 - 1.8 = 20.8$ inches
Upper limit: $\mu+2\sigma=22.6 + 2\times0.9=22.6+1.8 = 24.4$ inches

Step4: Calculate for 99.7% range

For approximately 99.7% of the data:
Lower limit: $\mu - 3\sigma=22.6-3\times0.9=22.6 - 2.7 = 19.9$ inches
Upper limit: $\mu+3\sigma=22.6+3\times0.9=22.6 + 2.7=25.3$ inches

Answer:

1. 21.7, 23.5
2. 20.8, 24.4
3. 19.9, 25.3