At a local manufacturing plant, employees must complete new machine set ups within 30 minutes. New machine set-up times can be described by a normal model with a mean of 22 minutes and a standard deviation of four minutes.
b. The typical worker needs five minutes to adjust to their surroundings before beginning their duties. What percent of new machine set-ups are completed within 25 minutes to allow for this?
77.3%
To find the percentage of new machine set-ups that are completed within 25 minutes, we need to calculate the z-score corresponding to 25 minutes and then find the corresponding area under the normal distribution curve.
Step 1: Calculate the z-score
The z-score is a measure of how many standard deviations a value is from the mean. It is calculated using the formula:
z = (x - μ) / σ
Where:
x is the value we want to find the z-score for (25 minutes in this case),
μ is the mean of the normal distribution (22 minutes in this case), and
σ is the standard deviation of the normal distribution (4 minutes in this case).
Plugging in the values, we get:
z = (25 - 22) / 4
z = 3 / 4
z = 0.75
Step 2: Find the area under the normal distribution curve
Once we have the z-score, we can look up the corresponding area under the normal distribution curve using a standard normal distribution table or a statistical software.
The area under the curve represents the percentage of set-ups completed within the given time frame. We can use a standard normal distribution table or a statistical software to find this area.
From a standard normal distribution table, we find that the area to the left of a z-score of 0.75 is approximately 0.7734.
Step 3: Convert the area to a percentage
To convert the area to a percentage, we multiply it by 100:
0.7734 * 100 = 77.34
Therefore, approximately 77.34% of new machine set-ups are completed within 25 minutes to allow for the 5-minute adjustment period.