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 solve this problem, we first need to find the total mass of the standard load of the crane when loading 8 cartons of sugar and 3 cartons of flour. Let \( X \) be the total mass of the load: \[ X = 8Y + 3Z \] where \( Y \) is the mass of a carton of sugar and \( Z \) is the mass of a carton of flour. From the problem, we have: - \( Y \sim N(78.8, 12.6^2) \) - \( Z \sim N(62.0, 10.0^2) \) To find the mean and variance of \( X \): 1. **Mean of \( X \)**: \[ E(X) = 8E(Y) + 3E(Z) = 8(78.8) + 3(62.0) = 630.4 + 186.0 = 816.4 \text{ kg} \] 2. **Variance of \( X \)**: Since \( Y \) and \( Z \) are independent: \[ Var(X) = 8^2 Var(Y) + 3^2 Var(Z) = 64(12.6^2) + 9(10.0^2) = 64(158.76) + 9(100) = 10160.64 + 900 = 11060.64 \] Thus, the standard deviation \( \sigma_X \) is: \[ \sigma_X = \sqrt{11060.64} \approx 105.16 \text{ kg} \] Now we can express \( X \) as a normal distribution: \[ X \sim N(816.4, 105.16^2) \] Next, we want to find the percentage of standard loads that exceed a maximum load of 900 kg. We will calculate the z-score for \( X = 900 \): \[ z = \frac{X - \mu}{\sigma} = \frac{900 - 816.4}{105.16} \approx \frac{83.6}{105.16} \approx 0.796 \] Now we will look up this z-score in the standard normal distribution table or use a calculator to find the area to the right of \( z = 0.796 \). The area to the left of \( z = 0.796 \) is approximately 0.7852. Therefore, the area to the right, which represents the percentage of standard loads exceeding 900 kg, is: \[ 1 - 0.7852 = 0.2148 \] Multiplying by 100 to get a percentage gives us: \[ \text{Percentage exceeding 900 kg} \approx 21.48\% \] So, about **21.5%** of standard loads will exceed the maximum safe load of 900 kg. **Assumption**: We assume the weights of the sugar and flour cartons are independent.