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.