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 can calculate the z-score of the strength requirement. The z-score formula is:
z = (x - μ) / σ
where:
x = strength requirement (4050 kg)
μ = mean breaking strength (4000 kg)
σ = standard deviation (25 kg)
Substituting the values into the formula:
z = (4050 - 4000) / 25
z = 50 / 25
z = 2
Now, we can find the proportion of products meeting the strength requirement by looking up the corresponding z-score in a standard normal distribution table. The table provides the proportion of values below a given z-score. In this case, we want to find the proportion above the z-score of 2, so we subtract the value from 1.
From the standard normal distribution table, we find that the proportion (p) below a z-score of 2 is approximately 0.9772. Therefore, the proportion above a z-score of 2 is:
p = 1 - 0.9772
p = 0.0228
This means that approximately 2.28% of products will not meet the strength requirement.
Since the manufacturer prefers that at least 95% of its products meet the strength requirement, we need to adjust the process parameter (mean). We need to find the z-score that corresponds to the desired proportion of 95%, which is 0.95. This value can be determined by finding the z-score in the standard normal distribution table.
From the standard normal distribution table, the z-score that corresponds to a proportion of 0.95 is approximately 1.645. Using the z-score formula:
1.645 = (x - 4000) / 25
Solving for x:
1.645 * 25 = x - 4000
41.125 = x - 4000
x = 4041.125
Therefore, if the manufacturer wants at least 95% of its products to meet the strength requirement, the process mean target should be set to 4041.125 kg.
To determine if the requirement is being met, we can calculate the probability of the cable's breaking strength being below the strength requirement of 4050 kg.
First, we need to convert the strength requirement into a standardized value, also known as the z-score. The formula for calculating the z-score is:
z = (x - μ) / σ
Where:
x = strength requirement (4050 kg)
μ = mean (4000 kg)
σ = standard deviation (25 kg)
Substituting the values, we get:
z = (4050 - 4000) / 25
= 50 / 25
= 2
Next, we need to find the probability of the cable's breaking strength being below the z-score of 2. We can use a standard normal distribution table or a statistical software to find this value. Looking up a standard normal distribution table, we find that the probability associated with a z-score of 2 is approximately 0.9772.
This implies that the probability of the cable's breaking strength being below 4050 kg is 0.9772. However, we are interested in the probability of it meeting the strength requirement, which is the complement of the probability of it being below 4050 kg. So, the probability of the cable's breaking strength meeting the requirement is approximately 1 - 0.9772 = 0.0228.
Since the probability of the cable's breaking strength meeting the requirement is only 0.0228 (or 2.28%), the requirement is not being met. Therefore, the process mean target needs to be adjusted to increase the breaking strength.
To determine the new process mean target, we can use the formula for the z-score and rearrange it to solve for the new mean (μ):
z = (x - μ) / σ
Solving for μ, we get:
μ = x - (z * σ)
= 4050 - (2 * 25)
= 4050 - 50
= 4000
Therefore, the new process mean target should be 4000 kg in order to ensure that at least 95% of the products meet the strength requirement of 4050 kg.