An employer wants to set a time limit so that 75% of the employees will finish a job on time. Past history has shown that the time required to do the job is normally distributed and has a mean time of 26 minutes with a standard deviation of 5 minutes. How much time should the employer allow employees to finish the job?

To find the time limit that will allow 75% of employees to finish the job on time, we can use the concept of z-scores.

Step 1: Calculate the z-score corresponding to the desired percentage. In this case, we want to find the z-score that corresponds to the 75th percentile. We can use a standard normal distribution table or a calculator to find this value. The z-score corresponding to the 75th percentile is approximately 0.674.

Step 2: Use the z-score formula to calculate the corresponding value in the original distribution. The formula for converting a z-score (Z) to an actual value (X) in a normal distribution is X = Z * standard deviation + mean.

Given the mean time of 26 minutes and standard deviation of 5 minutes, we can calculate the time limit as follows:

Time limit = Z * standard deviation + mean
= 0.674 * 5 + 26
≈ 3.37 + 26
≈ 29.37 minutes

Therefore, the employer should allow employees to finish the job within approximately 29.37 minutes.

To determine the time limit the employer should set, we need to find the duration that ensures 75% of employees will finish the job on time.

Since the job completion time is normally distributed with a known mean and standard deviation, we can use the properties of the normal distribution to find the appropriate value.

Step 1: Find the Z-score
To find the Z-score corresponding to the desired percentile (75%), we can use a standard normal distribution table or a statistical calculator. The Z-score represents the number of standard deviations a value is from the mean.

Step 2: Use the Z-score to find the duration
After obtaining the Z-score, we can use it to calculate the duration by applying the Z-score formula:

Z = (X - μ) / σ

where:
Z is the Z-score,
X is the value we want to find (duration),
μ is the mean of the distribution (26 minutes), and
σ is the standard deviation of the distribution (5 minutes).

Rearranging the formula to solve for X:

X = Z * σ + μ

By substituting the obtained Z-score, standard deviation, and mean into the equation, we can find the required duration.

Let's calculate it step by step.

Step 1: Finding the Z-score:
To find the Z-score that corresponds to the 75th percentile, we need to subtract the area to the left of it from 1.

1 - 0.75 = 0.25

Consulting a standard normal distribution table or using a statistical calculator, we find that the Z-score corresponding to an area of 0.25 is approximately 0.674.

Step 2: Calculating the duration:

X = Z * σ + μ
X = 0.674 * 5 + 26
X ≈ 29.37

Therefore, the employer should allow employees approximately 29.37 minutes to finish the job if they want 75% of them to complete it on time.