A sales contract between a manufacturer and a buyer requires 20 towels to be subjected to a water absorption test. If no more than 1 towels fails the test, the batch is accepted. If 2 or 3 towels fail the test, an additional 20 towels are tested. The batch is then accepted if 3 or less (out of 40) fail the test. Otherwise, the batch is rejected.
(a) If a batch of towels contains 5 % which would be rejected by the test,what is the probability that the batch is accepted.
(b) The lengths of 20 towels are measured and if the mean length is less than a value a specified in the contract, the batch is rejected. What should the value of a be to give a probability of 0.99 of accepting a batch with mean length of 106 mm and standard deviation of 6 mm.
I solved (a). The probability that the batch is accepted is 0.8961.
But I can't solve (b).
Here is my attempt on (b) :
Let X:length of a towel.
Y:total length of 20 towels
P( mean of Y >a ) = 0.99
P ( Y >20a ) = 0.99
P( Z > [20a-2120]/sqrt(720) ) = -2.326
[20a-2120]/sqrt(720) = 0.01
a = 102.9 mm
But the answer provided is 104.3 mm.
Can anyone help me with (b)? Thanks!