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 =  ~ N(0,1)
 ~ N(0,1)
where,  = population mean = 30.05 inches
 = population mean = 30.05 inches
             = standard deviation = 0.2 inches
   = 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(  <
 <  ) = P(Z < 3) = 0.99865
 ) = P(Z < 3) = 0.99865
  P(X bar <= 30.25) = P(  <=
 <=  ) = P(Z <= 2) = 0.97725
 ) = P(Z <= 2) = 0.97725
Therefore, P(30.25 inches < X bar < 30.35 inches)  = 0.99865 - 0.97725 
                                                                                        = 0.0214