The time necessary to assemble a discount store display is normally distributed with μ = 5 minutes and σ = 30 seconds. A large number of potential employees are timed assembling a practice display. How much time (in minutes) should the personnel manager allow the potential employees to assemble the display if the store wants only 80% of the people to complete the task?
a.4.86
b.5.0
c.5.42
d.5.89
To determine the amount of time the personnel manager should allow the potential employees to assemble the display, we need to find the corresponding value on the normal distribution curve.
Step 1: Convert the desired percentage (80%) to a corresponding z-score.
In this case, we want to find the z-score that corresponds to the lower 80% of the distribution since we want only 80% of the people to complete the task. We can use a standard normal distribution table or a calculator to find the z-score.
Step 2: Once we have the z-score, we can use the formula for z-score to find the corresponding value on the distribution.
The formula for z-score is: z = (X - μ) / σ
Where:
- z is the z-score
- X is the value on the distribution
- μ is the mean of the distribution
- σ is the standard deviation of the distribution
Step 3: Plug in the values into the formula and solve for X.
In this case:
- μ = 5 minutes
- σ = 30 seconds = 0.5 minutes
- z-score = the z-score we found in Step 1
Step 4: Convert the result back to minutes.
Given the answer choices provided, we can plug in the values and see which one matches the result.
Let's calculate the answer using this process:
Step 1: The z-score corresponding to the lower 80% is approximately 0.85, which we can find using a standard normal distribution table or a calculator.
Step 2: Plug in the values into the formula:
0.85 = (X - 5) / 0.5
Step 3: Solve for X:
0.85 * 0.5 = X - 5
0.425 = X - 5
X = 0.425 + 5
X = 5.425 minutes
Step 4: Convert the result back to minutes:
X ≈ 5.43 minutes
Therefore, the personnel manager should allow the potential employees approximately 5.43 minutes to assemble the display.
Looking at the answer choices, the closest option is c. 5.42 minutes.
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion (.8) in the largest portion and use the Z score to calculate the raw time (score). Don't forget to change to common units.