So my stats teacher isn't very motivating and I need to learn it better, however the work he sets, he never marks so i can never tell if i have done it right.
You answered Syed's Q on the same paper i believe.
Here's the Q:
Soup is sold in tins which are filled by a machine. The actual weight of soup delivered to a tin by the filling machine is always normally distributed about the mean weight with a standard deviation of 8g. The machine is set originally to deliver a mean weight of 810g.
(a) Determine the probability that the weight of soup in a tin, selected at random, is less than 800g.
(b) Determine the probability that the weight of soup in a tin, selected at random, is between 795 g and 820 g.
I understood all of this thanks to your answer,
But there's part (c) which wasn't mentioned:
Proposed legislation states that no more than 2.5 percent of tins may contain less than the nominal net weight of 800g
(c) Assuming the value of of the standard deviation remains unchanged, determine the minimum mean weight that the machine should be set to deliver in order to comply with this requirement.
What the question needs us to do is simply find the mean weight using the standardized (z) score corresponding to a weight of 800g such that the area to the left of it on the normal distribution curve is 2.5% or that the total area to the right of it is 97.5% or that the area to the right of it until the middle of the curve is 47.5%. They all mean the same thing and one of them would be the easiest depending on the table you're using.
From tables, this value can be checked to be
z = -1.96
z = w - μ / σ
μ = w - zσ
= 800 - (-1.96)(8)
= 800 + 15.68
and the minimum mean weight should be set to 815.68g
By doing this, the legal requirement can be ensured. If the mean weight is set at a higher value, then the percentage of tins that would contain less than 800g in an ideal normal distribution would even be lower than 2.5%, which is legal, but of course other production parameters would matter in a real world situation so it might not be feasible to set the mean weight at, say, 1000g just to satisfy a legality.