Question

chapter 3.2 homework
score: 51/102 answered: 20/33
question 21
3.2 measures of spread: empirical rule
the heights of women have a symmetric distribution with a mean of 67 inches and a standard deviation of 2.5 inches.

1. approximately 68% of women have heights between ( , ) inches.
2. approximately 95% of women have heights between ( , ) inches.
3. approximately 99.7% of women have heights between ( , ) inches.

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Explanation:

Step1: Recall empirical - rule for normal distribution

For a normal (symmetric) distribution, about 68% of the data lies within 1 standard - deviation of the mean, about 95% lies within 2 standard - deviations of the mean, and about 99.7% lies within 3 standard - deviations of the mean.

Step2: Calculate bounds for 68%

The mean $\mu = 67$ inches and the standard deviation $\sigma=2.5$ inches. For 68% of the data, the lower bound is $\mu-\sigma=67 - 2.5=64.5$ inches and the upper bound is $\mu+\sigma=67 + 2.5 = 69.5$ inches.

Step3: Calculate bounds for 95%

For 95% of the data, the lower bound is $\mu - 2\sigma=67-2\times2.5=67 - 5 = 62$ inches and the upper bound is $\mu+2\sigma=67 + 2\times2.5=67 + 5 = 72$ inches.

Step4: Calculate bounds for 99.7%

For 99.7% of the data, the lower bound is $\mu-3\sigma=67-3\times2.5=67 - 7.5 = 59.5$ inches and the upper bound is $\mu + 3\sigma=67+3\times2.5=67 + 7.5 = 74.5$ inches.

Answer:

1. 64.5, 69.5
2. 62, 72
3. 59.5, 74.5