Is there a more accurate way to estimate sales quantities?

[help]

Homework Statement

homework. Am I thinking right? modular inequality

Relevant Equations

The number of matchboxes that will be sold in a month can be estimated according to
the price p of each box. The estimated quantity of boxes sold is related
with the price p according to the equation

q = 16000 - 2500p
when the price is between 2 and 6.

The company cannot sell with a price below 2 and greater than or equal to 6.
Note that q is just an estimate of the quantity sold. In normal scenarios, this
estimate has a maximum error of 1,000 units in relation to the actual quantity determined
r at the end of the month.
Determine the price range that should be used for the number of boxes sold
is equal to 7,250 units, within the maximum error margin of the survey.



So, I started this problem and I would like to be sure of what I did.

attached resolution attempt

attached resolution attempt

 

[SammyS]

Homework Statement:: homework. Am I thinking right? modular inequality
Relevant Equations:: The number of matchboxes that will be sold in a month can be estimated according to
the price p of each box. The estimated quantity of boxes sold is related with the price p according to the equation

q = 16000 - 2500p
when the price is between 2 and 6.

The company cannot sell with a price below 2 and greater than or equal to 6.
Note that q is just an estimate of the quantity sold. In normal scenarios, this estimate has a maximum error of 1,000 units in relation to the actual quantity determined r at the end of the month. Determine the price range that should be used for the number of boxes sold is equal to 7,250 units, within the maximum error margin of the survey.

So, I started this problem and I would like to be sure of what I did.

attached resolution attempt

attached resolution attempt

Hello @help . (Maybe use a better name)


Your solution:

It looks correct to me.

 

[help]

It looks correct to me.

Hi, thanks for replying.
Letter Bb)
Over 6 months, several prices were tried. The following actual sales figures are observed, according to
with each price charged:

The company believes that a good estimate q of the quantity sold can be given by

Give the lowest value you can for E so that https://mathhelpforum.com/attachments/1587578280478-png.40117/ in all observations made

So, I would like to be sure of what I did.

p	q	r	E
3	3500	3250	250
3,2	3400	3570	170
3,5	3250	3100	150
4	3000	2890	110
4,5	2750	2550	200
5	2500	2500	0

 

[haruspex]

The company believes that a good estimate q of the quantity sold can be given by

Something missing here, like a new formula for q?

Give the lowest value you can for E so that

So that what?

 

[help]

Sorry!
The company believes that a good estimate q of the quantity sold can be given by
q = 5000 - 500p.

Give the lowest value you can for E so that |q - r | ≤ E in all observations made.

 

[haruspex]

Sorry!
The company believes that a good estimate q of the quantity sold can be given by
q = 5000 - 500p.

Give the lowest value you can for E so that |q - r | ≤ E in all observations made.

Then your work in post #3 looks correct, but what is your answer for the lowest value of E meeting the condition?

 

[help]

the lowest possible value for E would be zero correct?

to end the question has an item c...

c)
Calculate the percentage of error observed |q-r| in relation to an actual quantity sold each
month.
From there, your opinion on the quality of this estimate.

So, I would like to be sure of what I did.

percentage error = E/r

what does your opinion mean about the quality of this estimate?

p	q	r	E
3	3500	3250	250	7,69%​
3,2	3400	3570	170	4,76%​
3,5	3250	3100	150	4,84%​
4	3000	2890	110	3,81%​
4,5	2750	2550	200	7,84%​
5	2500	2500	0	0,00%​

 

[haruspex]

the lowest possible value for E would be zero correct?

No.
Your confusion comes from having used E in your table in a different way from the way it is used in the question.
Your table lists six values, so let's call these E_i, i=1..6.
The question is asking for the least value of another variable, E, such that |E_i|≤E for all i.

 

[help]

No.
Your confusion comes from having used E in your table in a different way from the way it is used in the question.
Your table lists six values, so let's call these E_i, i=1..6.
The question is asking for the least value of another variable, E, such that |E_i|≤E for all i.

it's clearing up a little bit. I can leave | q-3000 | ≤500 because so 2500≤q≤3500. That is E = 500 ?

p	q	r	r-q	|r-q|
3	3500	3250	-250	250
3,2	3400	3570	170	170
3,5	3250	3100	-150	150
4	3000	2890	-110	110
4,5	2750	2550	-200	200
5	2500	2500	0	0

 

[haruspex]

it's clearing up a little bit. I can leave | q-3000 | ≤500 because so 2500≤q≤3500. That is E = 500 ?

p	q	r	r-q	|r-q|
3	3500	3250	-250	250
3,2	3400	3570	170	170
3,5	3250	3100	-150	150
4	3000	2890	-110	110
4,5	2750	2550	-200	200
5	2500	2500	0	0

I don't understand how you get 500.
You have six values for |r-q|. What is the smallest number E such that none of the six values exceed E?

 

[help]

Oh ok I got it. It is a maximum error. then E = 250

 

[haruspex]

Oh ok I got it. It is a maximum error. then E = 250

Right.

 

[help]

Thanks!
to end the question has an item c...

c)
Calculate the percentage of error observed |q-r| in relation to an actual quantity sold each
month.
From there, your opinion on the quality of this estimate.

So, I would like to be sure of what I did.

percentage error = E/r

what does your opinion mean about the quality of this estimate?

p	q	r	r-q	|r-q|
3	3500	3250	-250	250	7,69%​
3,2	3400	3570	170	170	4,76%​
3,5	3250	3100	-150	150	4,84%​
4	3000	2890	-110	110	3,81%​
4,5	2750	2550	-200	200	7,84%​
5	2500	2500	0	0	0,00%​

 

[haruspex]

Thanks!
to end the question has an item c...

c)
Calculate the percentage of error observed |q-r| in relation to an actual quantity sold each
month.
From there, your opinion on the quality of this estimate.

So, I would like to be sure of what I did.

percentage error = E/r

what does your opinion mean about the quality of this estimate?

p	q	r	r-q	|r-q|
3	3500	3250	-250	250	7,69%​
3,2	3400	3570	170	170	4,76%​
3,5	3250	3100	-150	150	4,84%​
4	3000	2890	-110	110	3,81%​
4,5	2750	2550	-200	200	7,84%​
5	2500	2500	0	0	0,00%​

You have calculated the percentages correctly.
How good the estimate is depends on what it is to be used for and what the alternatives are.
Plot q and r against p. Does it suggest a better formula for q?