Question

The masses of packages from a particular machine are normally distributed with a mean of 200g and a standard deviation of 2g. Find the probability that a randomly selected package from the machine weighs:
Less than 196g
Between 198.5g and 199.5g

Answer 1

Answer on question #51649, Math, Statistics and Probability

Question The masses of packages from a particular machine are normally distributed with a mean of 200g and a standard deviation of 2g. Find the probability that a randomly selected package from the machine weighs: Less than 196g Between 198.5g and 199.5g

Solution So we have normal distribution here. Hence, to find probabilities we have to compute numerically integrals of type

$P(x_{1}<x<x_{2})\int_{x_{1}}^{x_{2}}\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x-\mu)^{2}}{2\sigma^{2}}}$

where $\mu$ is mean and $\sigma$ is standard deviation. We have $\mu=200$ and $\sigma=2$. For the first case $x_{1}=-\infty$, $x_{2}=196$. Hence

$P=\int_{-\infty}^{96}\frac{1}{2\sqrt{2\pi}}e^{-\frac{(x-200)^{2}}{2\cdot 2^{2}}}\approx 0.022750$

For the second case $x_{1}=198.5$, $x_{2}=199.5$. Hence

$P=\int_{198.5}^{199.5}\frac{1}{2\sqrt{2\pi}}e^{-\frac{(x-200)^{2}}{2\cdot 2^{2}}}\approx 0.174666$

The numerical computation was done using Wolfram Mathematica 6.0. The command is

Probability[198.5 $<$ x $<$ 199.5,x $\approx$ NormalDistribution[200, 2]]