Question

Acceptance sampling is an important quality control technique, where a batch of data is tested to determine if the proportion of units having a particular attribute exceeds a given percentage. Suppose that 11% of produced items are known to be nonconforming. Every week a batch of items is evaluated and the production machines are adjusted if the proportion of nonconforming items exceeds 15%. [You may find it useful to reference the z table.] a. What is the probability that the production machines will be adjusted if the batch consists of 52 items? (Round “z” value to 2 decimal places, and final answer to 4 decimal places.) b. What is the probability that the production machines will be adjusted if the batch consists of 104 items? (Round “z” value to 2 decimal places, and final answer to 4 decimal places.)

Answer 1

a) p = 0.11

    n = 52

= p = 0.11

= sqrt(p(1 - p)/n)

      = sqrt(0.11 * (1 - 0.11)/52)

      = 0.0434

P( > 0.15)

= P(( - )/ > (0.15 - )/)

= P(Z > (0.15 - 0.11)/0.0434)

= P(Z > 0.92)

= 1 - P(Z < 0.92)

= 1 - 0.8212

= 0.1788

b) p = 0.11

    n = 104

= p = 0.11

= sqrt(p(1 - p)/n)

      = sqrt(0.11 * (1 - 0.11)/104)

      = 0.0307

P( > 0.15)

= P(( - )/ > (0.15 - )/)

= P(Z > (0.15 - 0.11)/0.0307)

= P(Z > 1.30)

= 1 - P(Z < 1.30)

= 1 - 0.9032

= 0.0968