To solve these problems, we will use the standard normal distribution. Given that the project time follows a normal distribution with a mean of 44 weeks and a standard deviation of 8 weeks, we can standardize the values using the z-score formula:
z = (X - μ) / σ
where X is the given value, μ is the mean, and σ is the standard deviation.
a) To find the probability that the project is finished in 40 weeks or fewer, we need to calculate the area under the standard normal curve to the left of 40 weeks.
First, we calculate the z-score:
z = (40 - 44) / 8
z = -0.5
Next, we need to find the probability associated with this z-score. We can look up the value in the standard normal distribution table or use a calculator.
Using a table, we find that the area to the left of -0.5 is approximately 0.3085.
Therefore, the probability that the project is finished in 40 weeks or fewer is 0.3085, or 30.85%.
b) To find the probability that the project is finished in 52 weeks or fewer, we follow a similar process.
First, we calculate the z-score:
z = (52 - 44) / 8
z = 1.0
Using the standard normal distribution table or a calculator, we find that the area to the left of 1.0 is approximately 0.8413.
Therefore, the probability that the project is finished in 52 weeks or fewer is 0.8413, or 84.13%.
c) To find the number of weeks for which there is a 95% probability that the project will be finished, we need to determine the corresponding z-score.
Since we want to find the value that leaves 5% in the tail, we look up the z-score for a cumulative area of 0.95 in the standard normal distribution table. The z-score for 0.95 is approximately 1.645.
Now we use the standardized formula to solve for X, the number of weeks:
1.645 = (X - 44) / 8
Rearranging the equation, we find:
X - 44 = 1.645 * 8
X - 44 = 13.16
X = 57.16
Therefore, there is a 95% chance that the project will be finished in fewer than 57.16 weeks or approximately 57 weeks.