The breaking strength of cable is known to be normally distributtes with a mean of 4000 kg and a standard deviation of 25 kg.The manufacturer prefers that at least 95% of his product meet a strength requirement og 4050 kg.is that requirement being met?if not. by changing the process parameter,what should be the process mean target value be?
Using http://psych.colorado.edu/~mcclella/java/normal/accurateNormal.html ,
I find that only 2.3% of the cables will have a breaking strength of 4050 kg (force)
Assuming that the standard deviation remains the same, there should be a new target value of 4091 kg. Then, only 5% will have breaking strength below 4050
Can you please explain how did you come to this conclusion? I have gone to the site mentioned there but still am unable to understand more clearly.
To determine if the requirement is being met and to find the process mean target value, we can use z-scores and the standard normal distribution.
Step 1: Calculate the z-score for the desired strength requirement of 4050 kg.
The formula for z-score is:
z = (x - μ) / σ
Where:
x = desired 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
Step 2: Look up the z-score in the z-table.
The z-table shows the probability of obtaining a value less than or equal to the given z-score. In this case, we need to find the probability of obtaining a value less than or equal to 2.
By looking up the z-table, we find that the probability is approximately 0.9772. This represents the proportion of products meeting or exceeding the desired strength requirement.
Step 3: Compare the obtained probability with the desired probability.
The manufacturer prefers that at least 95% of the products meet the strength requirement. This means the desired probability is 0.95.
Since the obtained probability (0.9772) is greater than the desired probability (0.95), we can conclude that the requirement is being met.
However, if the requirement was not being met, we would need to adjust the process mean target value to increase the percentage of products meeting the strength requirement. To do this, we can use the inverse z-score formula:
x = μ + (z * σ)
Where:
x = new process mean target value
μ = current mean breaking strength
σ = standard deviation
z = desired z-score
Using the values given:
x = 4000 + (2 * 25)
x = 4000 + 50
x = 4050 kg
Therefore, if the requirement was not being met, the new process mean target value should be 4050 kg.
To determine if the requirement is being met, we need to calculate the probability that the breaking strength will be at least 4050 kg given the mean and standard deviation provided.
Let's assume the breaking strength follows a normal distribution with a mean (μ) of 4000 kg and a standard deviation (σ) of 25 kg. We are interested in finding the probability that the breaking strength is greater than or equal to 4050 kg.
1. Convert the requirement of 4050 kg to a z-score. The z-score formula is as follows: z = (x - μ) / σ, where x is the value we want to convert.
z = (4050 - 4000) / 25
z = 50 / 25
z = 2
2. Look up the z-score in the standard normal distribution table (also known as the z-table) to find the corresponding probability. The z-table provides the probability to the left of the given z-score.
From the z-table, we find that the probability to the left of z = 2 is approximately 0.9772.
3. Subtract this probability from 1 to find the probability that the breaking strength is greater than or equal to 4050 kg.
P(X >= 4050) = 1 - 0.9772
P(X >= 4050) = 0.0228
The probability that the breaking strength is greater than or equal to 4050 kg is approximately 0.0228 or 2.28%.
Since this probability is less than the desired 95%, the requirement is not being met. To adjust the process mean target value and increase the probability of meeting the requirement, we can use the z-score formula and solve for the desired mean (μ).
z = (4050 - μ) / 25
Rearranging the formula for μ, we get:
μ = 4050 - (z * 25)
Using the desired probability of 95% (corresponding to a z-score of approximately 1.645), we can calculate the adjusted process mean target value:
μ = 4050 - (1.645 * 25)
μ = 4050 - 41.125
μ ≈ 4008.875 kg
Therefore, the adjusted process mean target value should be approximately 4008.875 kg to ensure that at least 95% of the product meets the strength requirement of 4050 kg.