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. Between what two values (in ounces) symmetrically distributed around the population mean will 80 percent of the apples fall?
A. [2.13, 2.37]
B. [2.10, 2.40]
C. [2.06, 2.44]
D. [1.95, 2.55]

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

[tex]X \sim N(2.25,0.15)[/tex]

Where [tex]\mu=2.25[/tex] and [tex]\sigma=0.15[/tex]

For this case we want the limits for the middle 80% values of the distribution. so then we need 100-80= 20% of the area in the tails and 10% on each tail since the distribution is symmetrical.

We can use this condition for the lower limits

[tex]P(X>a)=0.9[/tex] (a)

[tex]P(X<a)=0.1[/tex] (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 0.1 of the area on the left and 0.9 of the area on the right it's z=-1.28. On this case P(Z<-1.28)=0.1 and P(z>-1.28)=0.9

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

[tex]P(X<a)=P(\frac{X-\mu}{\sigma}<\frac{a-\mu}{\sigma})=0.1[/tex]

[tex]P(z<\frac{a-\mu}{\sigma})=0.1[/tex]

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

[tex]z=-1.28<\frac{a-2.25}{0.15}[/tex]

And if we solve for a we got

[tex]a=2.25 -1.28*0.15=2.058[/tex]

So the value of height that separates the bottom 10% of data from the top 90% is 2.06.

For the upper limit since the distribution is symmetrical we can do this:

[tex]z=1.28<\frac{a-2.25}{0.15}[/tex]

And if we solve for a we got

[tex]a=2.25 +1.28*0.15=2.442[/tex]

So the value of height that separates the bottom 90% of data from the top 10% is 2.44.

And the best answer for this case would be:

C. [2.06, 2.44]