a. To find the probability that the production machines will be adjusted if the batch consists of 60 items, we need to find the probability that the proportion of nonconforming items exceeds 17%.
We can use the z-score formula to find the z-score corresponding to a proportion of 17%. The formula for the z-score is:
z = (p - P) / sqrt(P(1-P)/n)
Where p is the sample proportion (17% = 0.17), P is the population proportion (13% = 0.13), and n is the sample size (60).
Calculating the z-score:
z = (0.17 - 0.13) / sqrt(0.13(1-0.13)/60)
z = 0.04 / sqrt(0.13(0.87)/60)
z = 0.04 / sqrt(0.1121/60)
z = 0.04 / sqrt(0.0018683)
z = 0.04 / 0.0431661
z ≈ 0.9268
Using the z-table, we can find the probability associated with this z-score. Looking up z = 0.9268, we find that the corresponding probability is 0.8212.
Therefore, the probability that the production machines will be adjusted if the batch consists of 60 items is approximately 0.8212.
b. To find the probability that the production machines will be adjusted if the batch consists of 120 items, we follow the same steps as in part a, but with a sample size of 120.
Calculating the z-score:
z = (0.17 - 0.13) / sqrt(0.13(1-0.13)/120)
z = 0.04 / sqrt(0.13(0.87)/120)
z = 0.04 / sqrt(0.1121/120)
z = 0.04 / sqrt(0.0009342)
z = 0.04 / 0.0305367
z ≈ 1.3107
Using the z-table, we can find the probability associated with this z-score. Looking up z = 1.3107, we find that the corresponding probability is 0.9053.
Therefore, the probability that the production machines will be adjusted if the batch consists of 120 items is approximately 0.9053.
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 13% 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 17%. [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 60 items? (Round final answer to 4 decimal places.)
b. What is the probability that the production machines will be adjusted if the batch consists of 120 items? (Round final answer to 4 decimal places.)
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