Question

A filling machine puts an average of four ounces of coffee in jars, with a standard deviation of 0.25 ounces. forty jars filled by this machine are selected at random. what is the probability that the mean amount per jar filled in the sampled jars is less than 3.9 ounces

Answer 1

Let's attack this problem using the z-score concept.  The sample std. dev. here is (0.25 oz)/sqrt(40), or 0.040.  Thus, the z score representing 3.9 oz. is
        3.9 - 4.0
z = -------------- = -2.5
          0.040 

In one way or another we must find the area under the std. normal curve that lies to the left of z = -2.5.  Use a table of z-scores or a calculator with built-in statistics functions.  According to my TI-83 Plus calculator, that area is

0.006.  One way of interpreting this that with so small a standard deviation, most volumes of coffee put into the jars are very close to the mean, 4 oz.