Question

The lengths of pregnancies are normally distributed with a mean of 268 days and a standard deviation of 15 days. If 64 women are randomly selected, find the probability that they have a mean pregnancy between 266 days and 268 days. Round to four decimal places.

Answer 1

Answer:

0.3569 is the probability that they have a mean pregnancy between 266 days and 268 days.

Step-by-step explanation:

We are given the following information in the question:

Mean, μ =  268 days

Standard Deviation, σ =  15 days

We are given that the distribution of lengths of pregnancies is a bell shaped distribution that is a normal distribution.

Formula:

$z_{score} = \displaystyle\frac{x-\mu}{\sigma}$

Standard error due to sampling =

$\displaystyle\frac{\sigma}{\sqrt{n}} = \frac{15}{\sqrt{64}} = \frac{15}{8}$

P(pregnancy between 266 days and 268 days)

$P(266 \leq x \leq 268) = P(\displaystyle\frac{266 - 268}{\frac{15}{8}} \leq z \leq \displaystyle\frac{268-268}{\frac{15}{8}}) = P(-1.0667 \leq z \leq 0)\\\\= P(z \leq 0) - P(z < -1.067)\\= 0.5000 - 0.1431 = 0.3569 = 35.69\%$

$P(266 \leq x \leq 268) = 35.69\%$