A factory produces tools of which 98% are in good working order. Samples of 1000 tools are selected at random and tested. a) Find the mean and give it a practical interpretation. b) Find the standard deviation of the number of tools in good working order in these samples.

 A factory produces tools of which 98% are in good working order. Samples of 1000 tools are selected at random and tested.

a) Find the mean and give it a practical interpretation.
b) Find the standard deviation of the number of tools in good working order in these samples.

Solution

When a tool is selected, it is either in good working order with a probability of 0.98 or not in working order with a probability of 1 - 0.98 = 0.02.

When selecting a sample of 1000 tools at random, 1000 may be considered as the number of trials in a binomial experiment and therefore we are dealing with a binomial probability problem.

a) mean: μ=np=1000×0.98=980$\mu =np=1000\times 0.98=980$
In a sample of 1000 tools, we would expect that 980 tools are in good working order .

b) standard deviation: σ=n×p×(1−p)−−−−−−−−−−−−√=1000×0.98×(1−0.98)−−−−−−−−−−−−−−−−−−−√=4.43