Final answer:
The probability that the project will need more than 21 weeks to complete can be found by standardizing the deadline using a z-score calculation and then subtracting the value found from 1. A z-score of -3.67 indicates a high probability that the project will take longer than 21 weeks.
Step-by-step explanation:
To find the probability that the project will need more than 21 weeks to complete when it has an expected finish time of 32 weeks with a variance of 9 weeks (which implies a standard deviation of 3 weeks since the standard deviation is the square root of variance), we use the normal distribution. We first standardize the deadline time (21 weeks) using a z-score calculation.
The formula for the z-score is: Z = (X - μ) / σ
Where:
- X is the deadline time
- μ (mu) is the mean finish time
- σ (sigma) is the standard deviation
Plugging in the numbers, we get:
Z = (21 - 32) / 3
Z = -11 / 3
Z = -3.67
This z-score tells us how many standard deviations below the mean our deadline is. To find the probability of completing the project after this deadline, we look up the z-score in a standard normal distribution table or use a computational tool. This will give us the probability that a value is less than our z-score. Since we want the probability of being greater, we subtract this value from 1.
Typically, a z-score of -3.67 would give us a very high probability (close to 1) because 21 weeks is well below the mean. Therefore, it's almost certain that the project will take longer than 21 weeks.