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]

And if we solve for a we got

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:

And if we solve for a we got

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]

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

Where and

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

  (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 0.1 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:

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 10% of data from the top 90% is 2.06.

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

And if we solve for a we got

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]

what graph

step-by-step explanation:

ratios represent a pattern or relationship that can be scaled.

equivalent ratios are pairs of ratios that represent the same relationship between quantities, and can be generated by multiplying or dividing both quantities in the ratio by the same factor.

in this lesson, students will use double number lines and ratio reasoning to solve ratio problems. by partitioning double number lines into proportionate intervals, students will build an understanding of equivalent ratio relationships as those that are generated by multiplying or dividing two related quantities by the same number. this lesson will build upon students' prior understanding of multiplicative comparisons, number line reasoning and equivalent fractions (while highlighting the difference between equivalent fractions and equivalent ratios); and will build toward future work in this unit on building ratio tables, and work in future grades in proportionality, scaling, and expressing and graphing linear functions.

step-by-step explanation:

ratios represent a pattern or relationship that can be scaled.

equivalent ratios are pairs of ratios that represent the same relationship between quantities, and can be generated by multiplying or dividing both quantities in the ratio by the same factor.

in this lesson, students will use double number lines and ratio reasoning to solve ratio problems. by partitioning double number lines into proportionate intervals, students will build an understanding of equivalent ratio relationships as those that are generated by multiplying or dividing two related quantities by the same number. this lesson will build upon students' prior understanding of multiplicative comparisons, number line reasoning and equivalent fractions (while highlighting the difference between equivalent fractions and equivalent ratios); and will build toward future work in this unit on building ratio tables, and work in future grades in proportionality, scaling, and expressing and graphing linear functions.