Suppose that the length, in inches, of a witch's or wizard's wand is a normal random variable with parameters mu = 11 and sigma^2 = 2. (a) what percentage of wands are over 12 inches? leave you answer in terms of phi. (b) given that a wand is greater than 12 inches, what is the probability it is greater than or equal to 15 inches? leave you answer in terms of phi.

a)

b)

Step-by-step explanation:

1) 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".  The letter is used to denote the cumulative area for a b quantile on the normal standard distribution, or in other words:

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

Where and

2) Part a

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 in order to find these probabilities we can find tables for the normal standard distribution, excel or a calculator.  

3) Part b

For this case we want a conditional probability given by:

And in order to solve this probability we can use the Bayes theorem given by:

If we use this rule we have this:

The denominator is already founded on part a. So we need to find:

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 in order to find these probabilities we can find tables for the normal standard distribution, excel or a calculator.  

898.1666667

step-by-step explanation:

5389/6 = 898.1666667