Answer to Question #116626 in Statistics and Probability for Kyle Schutter

Question

What is the probability that a random sample of 70 pregnancies has a mean gestation period of 124 days or​ less?

Answer 1

Suppose the lengths of the pregnancies of a certain animal are approximately normally distributed with mean $\mu=120$ days and standard deviation $\sigma=10$ days. 

What is the probability that a random sample of 70 pregnancies has a mean gestation period of 124 days or​ less?

Let $\bar{X}=$ a mean gestation period: $\bar{X}\sim N(\mu, \sigma^2/n).$ Then

$$Z={\bar{X}-\mu\over \sigma/\sqrt{n}}\sim N(0, 1)$$

Given that $\mu=120, \sigma=10, n=70.$

$$P(\bar{X}\leq124)=P(Z<{124-120\over 10/\sqrt{70}})\approx$$

$$\approx P(Z\leq3.346640)\approx0.999591$$