Acceptance Sampling. Exercises 35 and 36 involve the method of acceptance sampling, whereby a shipment of a large number of items is accepted based on test results from a sample of the items.

AAA Batteries AAA batteries are made by companies including Duracell, Energizer, Eveready, and Panasonic. When purchasing bulk orders of AAA batteries, a toy manufacturer uses this acceptance sampling plan: Randomly select 50 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 2000 AAA 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?

It is given that a shipment of batteries is accepted if at most two batteries do not meet the specifications. The probability that a battery will not meet specifications is equal to 2%.

Let Xdenote the number of batteries that do not meet specifications.

Success is defined as getting a battery that does not meet specifications.

The probability of success is computed below:

$$\begin{gathered} p=2\% \\ =\frac{2}{100} \\ =0.02 \end{gathered}$$

The probability of failure is computed below:

$$\begin{gathered} q=1-p \\ =1-0.02 \\ =0.08 \end{gathered}$$

The number of trials (n) is equal to 50.

The binomial probability formula used to compute the given probability is as follows:

$P\left(X=x\right)={}^n{C}_x{\left(p\right)}^x{\left(q\right)}^{n-x}$

Using the binomial probability formula, the probability that at most two batteries do not meet specifications is computed below:

$$\begin{gathered} P\left(X⩽2\right)=P\left(X=0\right)+P\left(X=1\right)+P\left(X=2\right) \\ ={}^{50}{C}_0{\left(0.02\right)}^0{\left(0.08\right)}^{50-0}+{}^{50}{C}_1{\left(0.02\right)}^1{\left(0.08\right)}^{50-1}+{}^{50}{C}_2{\left(0.02\right)}^2{\left(0.08\right)}^{50-2} \\ =0.36417+0.371602+0.185801 \\ =0.922 \end{gathered}$$

Thus, the probability that the shipment of batteries will be accepted is equal to 0.922.

The probability suggests that 92% of such shipments will be accepted, and only 8% of the shipments will be rejected.

Since the probability of rejection of shipment is low, the quality of batteries produced is good.