On the graph at left, row 1 shows the discriminant (D) for the quadratic function.
Right now D=12.
On the next screen, you'll be able to change a, b, and
c. Before you do, make a prediction.
Which of these values can you change to make D
larger? How can you change that value?

b. [278.90, 288.55]

See the explanation below.

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

Where and

LOWER LIMIT

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 0.25 of the area on the left and 0.75 of the area on the right it's z=-0.674. On this case P(Z<-0.674)=0.25 and P(z>-0.674)=0.75

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 25% of data from the top 75% is 278.90.  

UPPER LIMIT

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 0.7 of the area on the left and 0.3 of the area on the right it's z=0.524. On this case P(Z<0.524)=0.7 and P(z>0.524)=0.3

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 70% of data from the top 30% is 289.78.  

With the procedure we got for the limits [278.9 , 289.78]

And since any of the options are on the list we take th most similar for this case:

b. [278.90, 288.55]