A machine is shut down for repairs if a random sample of 200 items selected from the daily output of the machine reveals at least 15% defectives. (Assume that the daily output has a large number of items.) If on a given day, the machine is producing only 10% defective items, what is the probability that it will be shut down? State and justify a condition that is required to apply a normal approximation to this binomial probability.
The number of defective ones from the sample of 200 items is modelled here as:
The distribution of the proportion here is given as:
The probability that the machine is shutdown is computed here as:
P( p >= 0.15)
Converting it to a standard normal variable, we have here:
Getting it from the standard normal tables, we have here:
Therefore 0.0092 is the required probability here.
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