To find the probability that Hopkins Hospitality will finish the project in 42 weeks or less, we need to follow a systematic approach using the given parameters and some statistical calculations outlined below.
First, we highlight the given data:
- Estimated time \(T_{CP}\) for the critical path is 42.5 weeks.
- Variance \(\sigma^2_{CP}\) along the critical path is 22.97 weeks.
- Target project completion time \(T\) is 42 weeks.
### Step-by-step Solution:
1. Calculate the Standard Deviation (\(\sigma_{CP}\)) of the Project Time:
The standard deviation is the square root of the variance. This helps in understanding the spread of the possible completion times around the mean.
[tex]\[
\sigma_{CP} = \sqrt{\sigma^2_{CP}} = \sqrt{22.97} \approx 4.793 \text{ weeks} \quad (\text{rounded to three decimal places})
\][/tex]
2. Calculate the Z-score:
The Z-score is a measure that describes how many standard deviations a particular value is from the mean. It is calculated as:
[tex]\[
Z = \frac{T - T_{CP}}{\sigma_{CP}}
\][/tex]
Substituting the given values, we get:
[tex]\[
Z = \frac{42 - 42.5}{4.793} \approx -0.104 \quad (\text{rounded to three decimal places})
\][/tex]
3. Determine the Probability:
The probability that the project will complete in 42 weeks or less can be found using the Cumulative Distribution Function (CDF) for the normal distribution at the computed Z-score.
Using a standard normal distribution table or a computational tool, we find the cumulative probability corresponding to the Z-score of -0.104.
[tex]\[
P(Z < -0.104) \approx 0.458 \quad (\text{rounded to three decimal places})
\][/tex]
### Conclusion:
The probability that Hopkins Hospitality will finish the project in 42 weeks or less is approximately:
[tex]\[
\boxed{0.458}
\][/tex]
