Question

When purchasing bulk orders of​ batteries, a toy manufacturer uses this acceptance sampling​ plan: Randomly select and test 54 batteries and determine whether each is within specifications. The entire shipment is accepted if at most 2 batteries do not meet specifications. A shipment contains 6000 ​batteries, and 2 ​% of them do not meet specifications. What is the probability that this whole shipment will be​ accepted? Will almost all such shipments be​ accepted, or will many be​ rejected?

a. The probability that this whole shipment will be accepted is nothing . ​(Round to four decimal places as​ needed.)

b. The company will accept nothing ​% of the shipments and will reject nothing ​% of the​ shipments, so ▼ many of the shipments will be rejected. almost all of the shipments will be accepted. ​(Round to two decimal places as​ needed.)

Answer 1

Ans:

a)Let x be the number of batteries which do not meet specifications,then x has binomial distribution with n=54 and p=0.02

P(accepted)=P(x<=2)

=P(x=0)+P(x=1)+P(x=2)

=(1-0.02)^54+⁵⁴C₁*0.02*(1-0.02)^53+⁵⁴C₂*0.02^2*(1-0.02)^52

=0.9063

The probability that this whole shipment will be accepted=0.9063

b)The company will accept nothing 90.63​% of the shipments and will reject nothing ​ 9.37% of the​ shipments, so almost all of the shipments will be accepted.​