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  

yes the total will be same because in both of them the expression.is rearranged but their sign remains the same hence it hss just been shuffled