burkinaval1b

2021-12-26

Assume that the duration of human pregnancies can be described by a Normal model with mean 266 days and standard deviation 16 days. a) What percentage of pregnancies should last between 270 and 280 days? b) At least how many days should the longest 25% of all pregnancies last? c) Suppose a certain obstetrician is currently providing prenatal care to 60 pregnant women. Let y̅ represent the mean length of their pregnancies. According to the Central Limit Theorem, what's the distribution of this sample mean, y̅? Specify the model, mean, and standard deviation. d) What's the probability that the mean duration of these patient's pregnancies will be less than 260 days?

enhebrevz

Expert

a) Soo we have  $\mu =266$  and  $\sigma =16$ . Hence,
$P\left(270\le X\le 280\right)=P\left(\frac{270-266}{16}\le z\le \frac{280-266}{16}\right)=P\left(0.25\le z\le 0.88\right)$
$P\left(0.25\le z\le 0.88\right)=P\left(z\le 0.88\right)-P\left(z\le 0.25\right)$ $=0.8106-0.5987$ $=0.2119$
So, percentage will be 21.1%
b) $P\left(Z\ge z\right)=0.25$
We know z=0.675
$\frac{x-266}{16}=0.675$
$x=276.8$
So andwer is 277 days.
c) We know $\mu =266$
$\sigma =\frac{16}{\sqrt{60}}=2.06$
d) $P\left(X<260\right)=P\left(z<\frac{260-266}{2.06}\right)=P\left(z<-2.914\right)=0.00187$