the price of a certain brand of printers is normally distributed with mean cost of $215 and standard…
the price of a certain brand of printers is normally distributed with mean cost of $215 and standard deviation $35.\na) what proportion of printers cost between $110 and $320?\nb) what proportion of printers cost less than $145?\nc) what proportion of printers cost more than $250?
Answer
Explanation:
Step1: Calculate z - scores
The z - score formula is $z=\frac{x-\mu}{\sigma}$, where $\mu = 215$ (mean), $\sigma = 35$ (standard deviation), and $x$ is the value from the data set.
Step2: For part a
For $x_1 = 110$, $z_1=\frac{110 - 215}{35}=\frac{- 105}{35}=-3$. For $x_2 = 320$, $z_2=\frac{320 - 215}{35}=\frac{105}{35}=3$. Using the standard normal distribution table, $P(-3<Z<3)=\Phi(3)-\Phi(-3)$. Since $\Phi(-z)=1 - \Phi(z)$, $P(-3<Z<3)=\Phi(3)-(1 - \Phi(3)) = 2\Phi(3)-1$. Looking up in the table, $\Phi(3)\approx0.9987$, so $P(-3<Z<3)=2\times0.9987 - 1=0.9974\approx99.74%$.
Step3: For part b
For $x = 145$, $z=\frac{145 - 215}{35}=\frac{-70}{35}=-2$. Using the standard - normal table, $P(Z < - 2)=1-\Phi(2)$. Looking up $\Phi(2)\approx0.9772$, so $P(Z < - 2)=1 - 0.9772 = 0.0228 = 2.28%$.
Step4: For part c
For $x = 250$, $z=\frac{250 - 215}{35}=\frac{35}{35}=1$. Using the standard - normal table, $P(Z>1)=1-\Phi(1)$. Looking up $\Phi(1)\approx0.8413$, so $P(Z > 1)=1 - 0.8413=0.1587 = 15.87%$.
Answer:
a) $99.74%$ b) $2.28%$ c) $15.87%$