If the number of defects found is more than one standard deviation above the mean, the manufacturing facility needs to recalibrate their machines. What is the probability that the facility needs to recalibrate their machines

Question

Braylon Conley

Mathematics

18 May, 19:10

If the number of defects found is more than one standard deviation above the mean, the manufacturing facility needs to recalibrate their machines. What is the probability that the facility needs to recalibrate their machines

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Simone Wu · Answer 1

16% probability that the facility needs to recalibrate their machines.

Step-by-step explanation:

We have to use the Empirical Rule to solve this problem.

Empirical Rule:

The Empirical Rule states that, for a normally distributed random variable:

68% of the measures are within 1 standard deviation of the mean.

95% of the measures are within 2 standard deviation of the mean.

99.7% of the measures are within 3 standard deviations of the mean.

What is the probability that the facility needs to recalibrate their machines?

They will have to recalibrate if the number of defects is more than one standard deviation above the mean.

We know that by the Empirical Rule, 68% of the measures are within 1 standard deviation of the mean. The other 100-68 = 32% is more than 1 standard deviation from the mean. Since the normal distribution is symmetric, of those 32%, 16% are more than one standard deviation below the mean, and 16% are more than one standard deviation above the mean.

So there is a 16% probability that the facility needs to recalibrate their machines.