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 z-score and find the corresponding probability.
Step 1: Calculate the z-score
The z-score formula is given by: z = (x - μ) / σ
where x is the value we want to find the probability for, μ is the mean, and σ is the standard deviation.
In this case, we want to find the z-score for a strength requirement of 4050 kg. Plugging in the values, we get:
z = (4050 - 4000) / 25
z = 50 / 25
z = 2
Step 2: Find the corresponding probability
We can use a standard normal distribution table or a calculator to find the probability associated with the z-score. The requirement is that at least 95% of the products meet the strength requirement.
Looking up the z-score of 2 in the standard normal distribution table, we find that the area under the curve is approximately 0.9772. This means that 97.72% of the products meet the strength requirement.
Since the probability is higher than 95%, the requirement is being met.
If the requirement was not being met, we would need to change the process parameter to increase the mean and shift the distribution to the right.
To find the new mean target that would meet the requirement, we can rearrange the z-score formula:
z = (x - μ) / σ
rearranging for x, we get:
x = z * σ + μ
In this case, we want to find the new mean target when z is the z-score for a desired probability of 95%. Looking up the z-score for 95% in the standard normal distribution table, we find it to be approximately 1.645.
Plugging in the values, we get:
x = 1.645 * 25 + 4000
x = 41.125 + 4000
x ≈ 4041.125 kg
Therefore, if the requirement was not being met, the new process mean target would need to be approximately 4041.125 kg for at least 95% of the products to meet the strength requirement.