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This problem involves a construction activity network for a house framing process, where the completion times of the activities are modeled as Gaussian random variables. Here's the approach to solve the problem:

### Problem Recap:

The activities involved in constructing the frame of a house are described in Table E4.9. Each activity has a mean and standard deviation for the completion time. The activities are assumed to be independent, and the framing cannot begin until both the foundation walls are completed and the components are delivered to the site. The goal is to find the probability that this can happen in 8 days or more.

### Steps:
1. **Determine the critical path:**
   - From the activity network, two paths lead to the completion of the framing activity:
     - Path 1: \( 1 \rightarrow 2 \rightarrow 3 \rightarrow 5 \)
     - Path 2: \( 1 \rightarrow 4 \rightarrow 5 \)
   - The time to complete each path is the sum of the activity durations along that path.

2. **Calculate the mean and standard deviation for each path:**
   - For Path 1: 
     \[
     \text{Activities: } 1 \rightarrow 2 \text{ (Excavation)}, 2 \rightarrow 3 \text{ (Construction of Footings)}, 3 \rightarrow 5 \text{ (Construction of Foundation Walls)}
     \]
     - Total mean: \( 1 + 2 + 3 = 6 \) days
     - Total standard deviation (since independent): 
       \[
       \sqrt{1^2 + 1^2 + 1^2} = \sqrt{3} \approx 1.732 \text{ days}
       \]
   - For Path 2:
     \[
     \text{Activities: } 1 \rightarrow 4 \text{ (Precutting and Subassembly)}, 4 \rightarrow 5 \text{ (Delivery of Components to Site)}
     \]
     - Total mean: \( 5 + 2 = 7 \) days
     - Total standard deviation:
       \[
       \sqrt{1^2 + \left(\frac{1}{2}\right)^2} = \sqrt{1 + 0.25} = \sqrt{1.25} \approx 1.118 \text{ days}
       \]

3. **Determine the critical path:**
   - Path 2 (with a mean of 7 days) takes longer than Path 1 (with a mean of 6 days), so Path 2 is the critical path.

4. **Probability that the framing will start in 8 days or more:**
   - The total time to complete Path 2 is a Gaussian random variable with mean \( \mu = 7 \) days and standard deviation \( \sigma = 1.118 \) days.
   - We want to find \( P(T \geq 8) \), where \( T \) is the completion time of Path 2.
   - Standardize the variable:
     \[
     Z = \frac{T - \mu}{\sigma} = \frac

The framing of a house may be done by subassembling the components in a plant and then delivering them to the site for framing. While this subassembly of components is being done, the preparation of the site, which includes the excavation and through construction of the foundation walls, can proceed at the same time. The activities may be represented with the activity network shown in Fig. E4.9 and described in Table E4.9. Assume that the completion time of each activity is a Gaussian random variable, with the respective means and standard deviations given. What is the probability that the framing can start in 8 days after the job starts?

Learn how to calculate the probability of starting the framing of a house in 8 days or more based on a construction activity network. The solution involves Gaussian random variables, the critical path method, and probability theory. This problem is ideal for undergraduate students in construction management and statistics.

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