Respuesta :
Answer:
Probability that the average length of a sheet is between 30.25 and 30.35 inches long is 0.0214 .
Step-by-step explanation:
We are given that 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.
Also, a sample of four metal sheets is randomly selected from a batch.
Let X bar = Average length of a sheet
The z score probability distribution for average length is given by;
Z = [tex]\frac{Xbar-\mu}{\frac{\sigma}{\sqrt{n} } }[/tex] ~ N(0,1)
where, [tex]\mu[/tex] = population mean = 30.05 inches
[tex]\sigma[/tex] = standard deviation = 0.2 inches
n = sample of sheets = 4
So, Probability that average length of a sheet is between 30.25 and 30.35 inches long is given by = P(30.25 inches < X bar < 30.35 inches)
P(30.25 inches < X bar < 30.35 inches) = P(X bar < 30.35) - P(X bar <= 30.25)
P(X bar < 30.35) = P( [tex]\frac{Xbar-\mu}{\frac{\sigma}{\sqrt{n} } }[/tex] < [tex]\frac{30.35-30.05}{\frac{0.2}{\sqrt{4} } }[/tex] ) = P(Z < 3) = 0.99865
P(X bar <= 30.25) = P( [tex]\frac{Xbar-\mu}{\frac{\sigma}{\sqrt{n} } }[/tex] <= [tex]\frac{30.25-30.05}{\frac{0.2}{\sqrt{4} } }[/tex] ) = P(Z <= 2) = 0.97725
Therefore, P(30.25 inches < X bar < 30.35 inches) = 0.99865 - 0.97725
= 0.0214
