Question

The population of lengths of aluminum-coated steel sheets is normally distributed with a mean of 30.05 inches and a standard deviation of .2 inches.

What is the probability that a randomly selected sample of 4 sheets will have an average length of less than 29.9 inches long?

Answer 1

Answer:  0.0668

Step-by-step explanation:

Given : The population of lengths of aluminum-coated steel sheets is normally distributed with a mean of 30.05 inches and a standard deviation of 0.2 inches.

i.e. $\mu=30.5$ and $\sigma=0.2$

Let x denotes the lengths of aluminum-coated steel sheets.

Then the probability that a randomly selected sample of 4 sheets will have an average length of less than 29.9 inches long will be :-

$P(x<29.9)=P(\dfrac{x-\mu}{\dfrac{\sigma}{\sqrt{n}}}<\dfrac{29.9-30.05}{\dfrac{0.2}{\sqrt{4}}})\\\\=P(z<-1.5)=1-P(z<1.5)\ \ [\because\ P(Z<-z)=1-P(Z<z)]$

$=1-0.9332= 0.0668$   [by using the z-value table ]

Hence, the required probability = 0.0668