An apple juice producer buys all his apples from a conglomerate of apple growers in one northwestern state. The amount of juice obtained from each of these apples is approximately normally distributed with a mean of 2.25 ounces and a standard deviation of 0.15 ounce. 77 percent of the apples will contain at least how many ounces of juice?

So the value of height that separates the bottom 23% of data from the top 77% is 2.139.

So we can conclude that 77% of the apples will contain at least 2.139 ounces  

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 heights of a population, and for this case we know the distribution for X is given by:

Where and

The z score formula is given by:

For this part we want to find a value a, such that we satisfy this condition:

  (a)

  (b)

Both conditions are equivalent on this case. We can use the z score again in order to find the value a.  

As we can see on the figure attached the z value that satisfy the condition with 0.23 of the area on the left and 0.77 of the area on the right it's z=-0.739. On this case P(Z<-0.739)=0.23 and P(z>-0.739)=0.7 7

If we use condition (b) from previous we have this:

But we know which value of z satisfy the previous equation so then we can do this:

And if we solve for a we got

So the value of height that separates the bottom 23% of data from the top 77% is 2.139.

So we can conclude that 77% of the apples will contain at least 2.139 ounces