Statistics and probability?

Question

In filling bags with flour, the automatic filling process dispenses amounts that follow a normal distrubution with a mean of 10.15kg and a standard deviation of .12kg. the label on the bags indicate that the amount of flour in the bags is 10kg.

(0) define a random variable (with a letter and in words) to be used in questions 1-3, and identify its probability distribution

(1) if you buy one bag of this flour, what is the probability that the bag contains at least as much flour as is promised on the label?

(2) what is the probability that the one bag you bough contains between 10.1 and 10.3kg?

(3) bags that are overfilled are prone to bursting. if a study suggests that 2percent of all bags are prone to bursting, what fill in kg do such bags have?

Answer 1

The nice thing about a normal distribution is that everything you need to know is contained in just two numbers: the mean and the standard deviation. That's partly because the normal distribution is symmetric, with the mean = median, etc.

In the normal distribution, the fraction of the population inside N standard deviations is given by:

erf(n/sqrt(2))

where erf is the error function. Here is a table of some values:

1 => 0.682689492137
2 => 0.954499736104
3 => 0.997300203937
4 => 0.999936657516
5 => 0.999999426697
6 => 0.999999998027

This means that 68.3% of the population is within 1 standard deviation of the mean, whatever it is, leaving the remaining 21.7% evenly split - half less than the median but more than one standard deviation away and half greater than the median.

This is all you need to know to solve your problems

So, for example, question 1. Let

B be the boundary you are looking for (in this case 10 kg)
M be the mean
S be the standard deviation

then N = (M - B)/S is the number of standard deviations between the boundary and the mean.

Next, find F = erf(N/sqrt(2)) using a table or a statistical calculator. Here is one source:
http://en.wikipedia.org/wiki/normal_dist...

F is the fraction that is within N standard deviations of the mean, but the remainder are split on either side of the mean. So:

O = 1-F is the total fraction outside the interval
H = O/2 is the fraction (= probability) of being outside the interval and below the mean.

In this case you want the central portion plus the portion above the mean so the fraction of the total is F + H

Similarly, you can go back and forth between standard deviations and kgs for the other two problems. (For question 2, since the mean is not 1.2 kg, you have two intervals around the mean to consider.)