Answer to Question #331080 in Statistics and Probability for Samaksh

Question

The reaction time of a driver to visual stimulus is normally distributed with a mean of 0.4

seconds and a standard deviation of 0.05 seconds. Use R to find the

(a) probability that a reaction requires more than 0.5 seconds.

(b) probability that a reaction requires between 0.4 and 0.5 seconds.

Answer 1

Denote by $X$ the random variable that has a normal distribution with parameters $\mu=0.4$ and $\sigma=0.05$. The probability density function is: $f(x)=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac12\left(\frac{x-\mu}{\sigma}\right)^2}$ .

(a) $P(X\geq0.5)=\int_{0.5}^{+\infty}\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac12\left(\frac{x-\mu}{\sigma}\right)^2}dx\approx0.0228$

(b) $P(0.4\leq X\leq0.5)=\int_{0.4}^{0.5}\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac12\left(\frac{x-\mu}{\sigma}\right)^2}dx\approx0.4772$

The answers are rounded to 4 decimal places