Toms of Maine sells toothpaste in a tube advertised to contain 8 ounces. In their continuous production process, the amount of toothpaste put in a tube is normally distributed with a mean of 8.21 ounces and a standard deviation of 0.09 ounces. If the actual capacity of the tubes used is 8.45 ounces, what proportion of the tubes will be filled beyond capacity?

And we can find this probability using the z table or excel:

Step-by-step explanation:

Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".  

Solution to the problem

Let X the random variable that represent the amount of toothpasye of a population, and for this case we know the distribution for X is given by:

Where and

The capacity is 8.45 ounces and we are interested on this probability

And the best way to solve this problem is using the normal standard distribution and the z score given by:

If we apply this formula to our probability we got this:

And we can find this probability using the z table or excel: