Answer to Question #336856 in Statistics and Probability for Lencho

Question

Suppose that a population of women heights is normally distributed with mean of 68

inches and standard deviation of 3 inches. If a person is selected at random, what is the

probability that her height is less than 66 inches?

Answer 1

Denote by $X$ a random variable, which is normally distributed with parameters $\mu=68$ and $\sigma=3$. The probability density function is: $p(x)=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac12\left(\frac{x-\mu}{\sigma}\right)^2}=\frac{1}{3\sqrt{2\pi}}e^{-\frac12\left(\frac{x-68}{3}\right)^2}$. The probability is: $P(X\leq66)=\int_{-\infty}^{66}\frac{1}{3\sqrt{2\pi}}e^{-\frac12\left(\frac{x-68}{3}\right)^2}dx\approx0.2525$. The answer is rounded to 4 decimal places.

Answer: $0.2525$ (rounded to 4 decimal places).