United Manufacturing and Supply makes sprinkler valves for u

United Manufacturing and Supply makes sprinkler valves for use in residential sprinkler systems. United supplies these valves to major companies such as Rain Bird and Nelson, who in turn sell sprinkler products to retailers. United recently entered into a contract to supply 40,000 sprinkler valves. The contract called for at least 97% of the valves to be free of defects. Before shipping the valves, United managers tested 200 randomly selected valves and found 190 defect-free valves in the sample. The managers wish to know the probability of finding 190 or fewer defect-free valves if in fact the population of 40,000 valves is 97% defect-free. The probability is:

Solution

We first get the z score for the critical value:          
          
x = critical value =    190      
u = mean = np =    194      
          
s = standard deviation = sqrt(np(1-p)) =    2.412467616      
          
Thus, the corresponding z score is          
          
z = (x-u)/s =    -1.658053345      
          
Thus, the left tailed area is          
          
P(z <   -1.658053345   ) =    0.0475 [ANSWER, C]

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[Actually, if we have a continuity correction adding 0.5 to 190, the answer will be 0.073418243, which is not in the choices.]