The driver of a truck loaded with 900 boxes of books will be fined if the total weight of the boxes exceeds 36450 pound. If the distribution of the weight of a box has a mean of 40 pounds and a variance of 36, find the approximate probability that the driver will be fined.

To solve this problem, we need to use the concept of the Central Limit Theorem (CLT) and the standard normal distribution.

The Central Limit Theorem states that for a large enough sample size, the distribution of the sample means will approach a normal distribution, regardless of the shape of the original population distribution.

We can calculate the mean and standard deviation of the total weight of the boxes using the properties of the normal distribution.

Step 1: Calculate the mean and standard deviation of the total weight of the boxes.

The mean of the total weight can be calculated by multiplying the mean of a single box by the number of boxes:
Mean = 40 pounds/box * 900 boxes = 36000 pounds.

The variance of the total weight can be calculated by multiplying the variance of a single box by the number of boxes:
Variance = 36 pounds^2/box * 900 boxes = 32400 pounds^2.

The standard deviation of the total weight is the square root of the variance:
Standard Deviation = √(32400 pounds^2) = 180 pounds.

Step 2: Convert the problem to a standard normal distribution.

To find the probability, we need to convert the problem into a standard normal distribution. We need to calculate the z-score, which measures how many standard deviations an observation or value is from the mean.

z = (x - μ) / σ.

In this case, x represents the weight threshold for being fined (36450 pounds), μ represents the mean of the total weight (36000 pounds), and σ represents the standard deviation of the total weight (180 pounds).

z = (36450 - 36000) / 180 = 2.5.

Step 3: Find the probability using the standard normal distribution table or calculator.

The final step is to find the probability using the z-score. Since we are interested in the probability that the total weight exceeds the threshold for being fined, we need to calculate the area to the right of the z-score.

P(Z > 2.5) = 1 - P(Z < 2.5).

Using a standard normal distribution table or calculator, we can find that P(Z < 2.5) is approximately 0.9938. Therefore,

P(Z > 2.5) ≈ 1 - 0.9938 = 0.0062.

So, the approximate probability that the driver will be fined is 0.0062, or 0.62%.