Answer:
0.0026 = 0.26% probability that the manufacturing line will be shut down unnecessarily
Step-by-step explanation:
We need to understand the normal probability distribution and the central limit theorem to solve this question.
Normal probability distribution
Problems of normally distributed samples are solved using the z-score formula.
In a set with mean and standard deviation
, the zscore of a measure X is given by:
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
Central Limit Theorem
The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean and standard deviation
, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean
and standard deviation
.
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
In this problem, we have that:
What is the probability that the manufacturing line will be shut down unnecessarily?
Less than 0.73 or more than 0.77.
Less than 0.73
pvalue of Z when X = 0.73
By the Central Limit Theorem
has a pvalue of 0.0013
More than 0.77
has a pvalue of 0.9987
1 - 0.9987 = 0.0013
2*0.0013 = 0.0026
0.0026 = 0.26% probability that the manufacturing line will be shut down unnecessarily