The time required is known to complete a project is known to be normally distributed with a mean of 44 weeks and standard deviations of 8 weeks
a) what is the probability that the project is finished in 40 weeks or fewer?
b) what is the probability that the project is finished in 52 weeks or fewer?
c) there is an 95 percentage chance that the project will be finished in fewer than how many weeks?
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.
To find the probabilities, we can use the Z-score formula and the standard normal distribution.
a) Probability of finishing the project in 40 weeks or fewer:
To find this probability, we need to calculate the Z-score for 40 weeks and then find the corresponding probability from the standard normal distribution.
Z = (X - μ) / σ
Z = (40 - 44) / 8
Z = -0.5
We can look up the Z-score of -0.5 in the standard normal distribution table or use a calculator to find the corresponding probability. The probability of finishing the project in 40 weeks or fewer is approximately 0.3085 or 30.85%.
b) Probability of finishing the project in 52 weeks or fewer:
Again, we need to calculate the Z-score for 52 weeks and find the corresponding probability.
Z = (X - μ) / σ
Z = (52 - 44) / 8
Z = 1
Using the Z-score table or a calculator, the probability of finishing the project in 52 weeks or fewer is approximately 0.8413 or 84.13%.
c) 95% chance of finishing in fewer than how many weeks:
To find the number of weeks that corresponds to a 95% probability, we need to find the Z-score for this probability and then solve for X.
Using the Z-score table or calculator, we find that the Z-score associated with a 95% probability is approximately 1.645.
Z = (X - μ) / σ
1.645 = (X - 44) / 8
Solving for X:
1.645 * 8 = X - 44
X = 13.16 + 44
X ≈ 57.16
There is a 95% chance that the project will be finished in fewer than approximately 57.16 weeks.