Final answer:
Using the Z-score for a 20% likelihood in a normal distribution and the given mean and variance, the project has a 20% chance of completion before day 112. The calculation involves converting the Z-score to the corresponding time.
Step-by-step explanation:
To determine the day by which there is a 20% chance that the project will be completed, we can use the concept of the normal distribution and Z-scores because the distribution of project completion times is often modeled as normal.
The Z-score for the given percentage can be found using the standard normal distribution table or a Z-score calculator. A 20% chance corresponds to a Z-score of approximately -0.84 (looking up the value where the left tail of the distribution includes 20% of the area under the curve).
Once the Z-score is identified, we use the following formula to find the time associated with this Z-score:
Z = (X - μ) / σ
Where:
• Z is the Z-score (-0.84),
• X is the unknown time (the day we want to find),
• μ is the mean (the estimated time of 120 days),
• and σ is the standard deviation (the square root of the variance, which is √100, or 10 days).
Plugging the values in, we get:
-0.84 = (X - 120) / 10
X = 120 + (-0.84 * 10)
X = 120 - 8.4
X = 111.6
Since we need to round to the nearest day, the project has a 20% chance of being completed before day 112, which corresponds to the options provided.