Probability: Normal Distribution Problem.

Question

The mass of waste in filled large dustbin bags is normally distributed with a mean of 6.8 kg and a standard deviation of 1.5 kg. The mass of waste in filled small dustbin bags is normally distributed with a mean of 3.2 kg and a standard deviation of 0.6 kg. One week there are 8 large and 3 small dustbin bags left for collection outside a block of flats. Find the probability that this waste has a total mass of more than 70 kg.

Answer 1

Large bags mean is 6.8 and SD is 1.5. For 8 bags we are multiplying both by 8, so the mean is 54.4 and SD is 12.

Small bags mean is 3.2 and SD is 0.6. For 3 bags we are multiplying both by 3, so the mean is 9.6 and SD is 1.8.

To find mean of entire set of bags, we add mean, so mean is 54.4 + 9.6 = 64

To find SD of entire set of bags, we take the square root of the sum of the variances= sqrt( 12^2 + 1.8^2) = 12.62

Let x = total mass, we want to find P(x > 70), we will use z-statistic to transform x to standard normal distribution.

z = (x - mean)/SD

P(x > 70) = P( z > (70 - 64)/12.62) = P(z > 0.48) = 1 - P(z < 0.48) = 1 - 0.6844 = 0.3156