Question

suppose that the lengths of human pregnancies are normally distributed with a mean of 268 days and a standard deviation of 14 days. complete the following statements.
(a) approximately 68% of pregnancies have lengths between days and days.
(b) approximately of pregnancies have lengths between 240 days and 296 days.

Explanation:

Step1: Recall the empirical rule for part (a)

The empirical rule states that for a normal distribution, approximately \(68\%\) of the data lies within \(1\) standard deviation (\(\sigma\)) of the mean (\(\mu\)).
Given \(\mu = 268\) and \(\sigma=14\).
The lower bound is \(\mu-\sigma=268 - 14\)

$$268-14=254$$

The upper bound is \(\mu+\sigma=268 + 14\)

$$268 + 14=282$$

Step2: Calculate the number of standard deviations for part (b)

For the lower bound \(x_1 = 240\), the \(z -\)score is \(z_1=\frac{x_1-\mu}{\sigma}=\frac{240 - 268}{14}=\frac{- 28}{14}=-2\)
For the upper bound \(x_2 = 296\), the \(z -\)score is \(z_2=\frac{x_2-\mu}{\sigma}=\frac{296 - 268}{14}=\frac{28}{14}=2\)
The empirical rule states that approximately \(95\%\) of the data lies within \(2\) standard deviations of the mean.

Answer:

(a) \(254\) days and \(282\) days.
(b) \(95\%\)