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.
a. What percent of new machine set ups take more than 30 minutes?
2.28%
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion related to that Z score.
To find the percent of new machine set ups that take more than 30 minutes, we need to calculate the area under the normal curve to the right of 30 minutes.
Step 1: Standardize the value
We need to find the z-score for 30 minutes using the formula:
z = (x - μ) / σ
where x is the value, μ is the mean, and σ is the standard deviation.
z = (30 - 22) / 4
z = 8 / 4
z = 2
Step 2: Find the probability
Next, we need to find the probability associated with a z-score of 2 using a standard normal distribution table or a calculator. The probability associated with a z-score of 2 is approximately 0.9772.
Step 3: Convert to percentage
Since the question asks for the percentage, we multiply the probability by 100 to get the answer in percent.
Percentage = 0.9772 * 100
Percentage ≈ 97.72%
Therefore, approximately 97.72% of new machine set ups take more than 30 minutes.
To find the percentage of new machine set-ups that take more than 30 minutes, we need to calculate the area to the right of 30 minutes in the normal distribution.
Step 1: Standardize the value
To do this, we will use the formula for standardizing a value in a normal distribution:
z = (x - μ) / σ
where:
- x is the value we want to standardize (in this case, 30 minutes),
- μ is the mean of the distribution (22 minutes), and
- σ is the standard deviation (4 minutes).
Substituting the values into the formula, we get:
z = (30 - 22) / 4
z = 8 / 4
z = 2
Step 2: Find the area under the standard normal curve
We can use a standard normal distribution table or a statistical calculator to find the area to the right of z = 2. Alternatively, we can use a calculator or software to calculate the cumulative probability directly.
Using a calculator or software, we find that the area to the right of z = 2 is approximately 0.0228.
Step 3: Convert the area to a percentage
To get the percentage, we can multiply the area by 100:
Percentage = 0.0228 * 100
Percentage ≈ 2.28%
Therefore, approximately 2.28% of new machine set-ups take more than 30 minutes.