3) The masses, in kilograms, of cartons of sugar and cartons of flour have the distributions \( N\left(78.8,12.6^{2}\right) \) and \( N\left(62.0,10.0^{2}\right) \) respectively. a) The standard load for a certain crane is 8 cartons of sugar and 3 cartons of flour. The maximum load that can be carried safely by the crane is 900 kg . Stating a necessary assumption, find the percentage of standard loads that will exceed the maximum safe load. [5]

To determine the percentage of standard loads that exceed the maximum safe load of 900 kg, we'll follow these steps: ### **a) Assumptions and Calculations** **Assumption:** We assume that the masses of cartons of sugar (**S**) and flour (**F**) are **independent**. **Given Distributions:** - **Sugar (S):** \( S \sim N(78.8\, \text{kg}, 12.6^2\, \text{kg}^2) \) - **Flour (F):** \( F \sim N(62.0\, \text{kg}, 10.0^2\, \text{kg}^2) \) **Standard Load Composition:** - **8 cartons of sugar:** \( 8S \) - **3 cartons of flour:** \( 3F \) - **Total Load (T):** \( T = 8S + 3F \) ### **Step 1: Calculate the Mean of Total Load (T)** \[ \begin{align*} E[8S] &= 8 \times 78.8 = 630.4\, \text{kg} \\ E[3F] &= 3 \times 62.0 = 186.0\, \text{kg} \\ E[T] &= 630.4 + 186.0 = 816.4\, \text{kg} \end{align*} \] ### **Step 2: Calculate the Variance and Standard Deviation of Total Load (T)** Since **S** and **F** are independent: \[ \begin{align*} \text{Var}[8S] &= 8^2 \times 12.6^2 = 64 \times 158.76 = 10,160.64\, \text{kg}^2 \\ \text{Var}[3F] &= 3^2 \times 10.0^2 = 9 \times 100 = 900\, \text{kg}^2 \\ \text{Var}[T] &= 10,160.64 + 900 = 11,060.64\, \text{kg}^2 \\ \text{SD}[T] &= \sqrt{11,060.64} \approx 105.17\, \text{kg} \end{align*} \] ### **Step 3: Determine the Z-Score** We want to find the probability that \( T > 900 \) kg. \[ Z = \frac{900 - E[T]}{\text{SD}[T]} = \frac{900 - 816.4}{105.17} \approx \frac{83.6}{105.17} \approx 0.796 \] ### **Step 4: Find the Probability Corresponding to the Z-Score** Using standard normal distribution tables: \[ P(Z > 0.796) \approx 1 - \Phi(0.796) \approx 1 - 0.7869 = 0.2131 \] ### **Conclusion** **Approximately 21%** of standard loads will exceed the maximum safe load of 900 kg. **Final Answer:** Assuming carton weights are independent, about 21 % of standard loads will surpass 900 kg.