Question

the lengths of a particular animals pregnancies are approximately normally distributed, with mean \\( \mu = 261 \\) days and standard deviation \\( \sigma = 16 \\) days.
(a) what proportion of pregnancies lasts more than 281 days?
(b) what proportion of pregnancies lasts between 253 and 269 days?
(c) what is the probability that a randomly selected pregnancy lasts no more than 241 days?
(d) a \very preterm\ baby is one whose gestation period is less than 221 days. are very preterm babies unusual?
(a) the proportion of pregnancies that last more than 281 days is \\( \square \\).
(round to four decimal places as needed.)

Explanation:

Step1: Calculate the z - score

The formula for the z - score is \(z=\frac{x-\mu}{\sigma}\). Here, \(\mu = 261\), \(\sigma=16\), and \(x = 281\).

$$z=\frac{281 - 261}{16}=\frac{20}{16}=1.25$$

Step2: Find the proportion

We want \(P(X>281)\), which is equivalent to \(P(Z > 1.25)\) since \(X\sim N(\mu = 261,\sigma = 16)\) and \(Z=\frac{X-\mu}{\sigma}\sim N(0,1)\).
Using the property \(P(Z>z)=1 - P(Z\leq z)\), and from the standard normal table \(P(Z\leq1.25)=0.8944\)

$$P(Z > 1.25)=1-0.8944 = 0.1056$$

Answer:

\(0.1056\)