Question

Lengths of pregnancies of humans are normally distributed with a mean of 267 days and a standard deviation of 14 days. Use the Empirical Rule to determine the percentage of women whose pregnancies are between 265 and 295 days.

Answer 1

Answer on Question # 32118 – Math – Statistic and Probability

Lengths of pregnancies of humans are normally distributed with a mean of 267 days and a standard deviation of 14 days. Use the Empirical Rule to determine the percentage of women whose pregnancies are between 265 and 295 days.

Solution.

Step 1: Sketch the curve.

The probability that $265 < X < 295$ is equal to the blue area under the curve.

Step 2:

Since $\mu = 267$ and $\sigma = 14$ we have:

$$P \left(265 < X < 295\right) = P \left(265 - 267 < X - \mu < 295 - 267\right) = P \left(\frac{265 - 267}{14} < \frac{X - \mu}{\sigma} < \frac{295 - 267}{14}\right)$$

Since $Z = \frac{\sigma - \mu}{\sigma} \cdot \frac{265 - 267}{14} = -0.14$ and $\frac{295 - 267}{14} = 2$ we have:

$$P \left(265 < X < 295\right) = P \left(-0.14 < Z < 2\right)$$

Step 3: Use the standard normal table to conclude that

$$P \left(-0.14 < Z < 2\right) = 0.5329$$

Answer: 53.3%

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