stats maths help

Question

Cornflakes are packed in boxes and the boxes are then packed into cartons.
Each carton contains 10 boxes. The weights of the boxes of cornflakes are
normally distributed with a mean of 500g and a standard deviation of 25g.
The weights of the empty cartons are also normally distributed with mean = 2000g
and  standard deviation = 100g.

90% of full cartons lie below a certain weight (x). Calculate the value of
x.

Answer 1

Box weight is 500g+25g, so 10 boxes weigh 5000g+250g. Carton weight is 2000g+100g. Full carton weight=7000g+350g. The standard deviations have simply been added together to give the worst case deviation.

The z-score corresponding to 90% in normal distribution is 1.28, so 1.28*350=448g deviation from the mean of 7000g. Therefore, x=7448g, or 7.448kg is the weight below which 90% of the full cartons can be expected to weigh.

Perhaps a better way of combining the standard deviations would be to go for the RMS value based on the percentage of the mean of the standard deviation; so 250/5000=5%=100/2000. The RMS value is (sqrt(2*0.05^2))/2=0.025sqrt(2)=0.3536. Converted to weight this is 7000*0.03536=247.5g. In this case, 1.28*247.5=316.78g, so 90% of full cartons can be expected to weigh below 7247.5g approx.