Question

suppose that the lengths of human pregnancies are normally distributed with a mean of 266 days and a standard deviation of 14 days. complete the following statements.
(a) approximately 99.7% of pregnancies have lengths between days and days.
(b) approximately of pregnancies have lengths between 252 days and 280 days.

Explanation:

Step1: Recall the empirical rule for normal distribution

The empirical rule states that for a normal distribution:

• Approximately \(68\%\) of the data lies within \(1\) standard deviation (\(\mu\pm\sigma\)) of the mean.
• Approximately \(95\%\) of the data lies within \(2\) standard deviations (\(\mu\pm2\sigma\)) of the mean.
• Approximately \(99.7\%\) of the data lies within \(3\) standard deviations (\(\mu\pm3\sigma\)) of the mean.

Step2: Solve part (a)

Given \(\mu = 266\) and \(\sigma=14\).
For \(99.7\%\) of the data (within \(3\) standard deviations):
Lower bound: \(\mu - 3\sigma=266-3\times14=266 - 42=224\)
Upper bound: \(\mu + 3\sigma=266+3\times14=266 + 42=308\)

Step3: Solve part (b)

First, find the number of standard deviations from the mean.
For \(x = 252\): \(z=\frac{252 - 266}{14}=\frac{- 14}{14}=-1\)
For \(x = 280\): \(z=\frac{280 - 266}{14}=\frac{14}{14}=1\)
Since \(z=-1\) and \(z = 1\), by the empirical rule, approximately \(68\%\) of the data lies within \(1\) standard deviation of the mean.

Answer:

(a) \(224\) days and \(308\) days.
(b) \(68\%\)