Question

1. a machine produces bottles with volume (mu = 500ml,sigma = 5ml). find (p(x < 495)).

(z=\frac{495 - 500}{5}=\frac{-5}{5}=-1) normalcdf((-1e99,-1,0,1)=0.1587)

1. find (p(490 < x < 505))

Explanation:

Step1: Calculate the z - scores

For \(x = 490\), \(z_1=\frac{490 - 500}{5}=\frac{- 10}{5}=-2\)
For \(x = 505\), \(z_2=\frac{505 - 500}{5}=\frac{5}{5}=1\)

Step2: Use the normalcdf function

We use the formula \(P(a<X<b)=\text{normalcdf}(a,b,\mu,\sigma)\) or for the standard normal distribution \(P(z_1<Z<z_2)=\text{normalcdf}(z_1,z_2,0,1)\)
\(P(490 < X < 505)=\text{normalcdf}(-2,1,0,1)\)

Using a calculator or standard normal table:
\(\text{normalcdf}(-2,1,0,1)=\Phi(1)-\Phi(-2)\)
where \(\Phi(z)\) is the cumulative distribution function of the standard normal distribution.
\(\Phi(1) = 0.8413\), \(\Phi(-2)=0.0228\)
\(P(490 < X < 505)=0.8413-0.0228 = 0.8185\)

Answer:

\(0.8185\)