the breaking strength of a cable known to be normally distributed with a mean of 4000kg and a standard deviation of 25kg.The manufacture prefers that at least 95% of its products meet a strength requirement of 4050 kg.Is this requirement being met? If not, by changing the process parameter what would the process mean target would be?
To determine if the requirement is being met, we need to calculate the probability that the breaking strength is less than 4050 kg.
Here's how you can calculate the probability using the normal distribution:
1. Calculate the Z-score:
Z = (X - μ) / σ
Where X is the desired breaking strength (4050 kg), μ is the mean breaking strength (4000 kg), and σ is the standard deviation (25 kg).
Z = (4050 - 4000) / 25
2. Lookup the probability for the Z-score in the Z-table. The Z-table provides the area under the normal distribution curve for a given Z-score.
The resulting Z-score is 2. This means that we want to find the probability of a breaking strength below 4050 kg, which is the area to the left of the Z-score of 2.
3. Find the probability using the Z-table:
The Z-score of 2 corresponds to a probability of 0.9772.
So, the probability of a breaking strength below 4050 kg is 0.9772.
Since the probability is above 0.95, it implies that more than 95% of the products are meeting the strength requirement of 4050 kg. Therefore, the requirement is being met.
If the requirement were not being met, and the manufacturer wanted to change the process parameters to ensure that at least 95% of the products meet the strength requirement, they would need to adjust the process mean target.
To find the new process mean target, we would need to find the value of X that corresponds to a desired probability of 0.95. This can be done by using the inverse of the normal distribution (also known as the Z-inverse or Z-score).
1. Look up the value of Z-inverse for a desired probability of 0.95. In this case, it is typically denoted as Zα/2, where α is the complement of the desired probability (1 - 0.95 = 0.05) and divided by 2 to get the area under both tails of the distribution.
Zα/2 = Z(0.05/2) = Z(0.025)
2. Find the Z-inverse in the Z-table. The resulting Z-inverse is approximately 1.96.
3. Calculate the new process mean target:
X = Z * σ + μ
Where Z is the Z-inverse (1.96), σ is the standard deviation (25 kg), and μ is the current mean (4000 kg).
X = 1.96 * 25 + 4000
The new process mean target would be approximately 4050.
So, if the requirement were not being met, the manufacturer would need to adjust the process mean target to approximately 4050 kg in order to ensure that at least 95% of the products meet the strength requirement.