Question

Day 1 2 3 4 5 6
Cans 50 60 70 75 55 ?

Megan is collecting cans for a food drive. The number of cans she has collected so far is shown in the table. Which number of cans collected on day 6 will give Megan’s data a mean absolute deviation of 8.3?
A) 50
B) 56
C) 60
D) 66

Answer 1

I'm not sure but I think it's 50. Basically, for MAD, if I recall correctly, you have to find the average of your input then find the average of the distance of each entry from the average (hence the "absolute"). Can you try it out?

Answer 2

I got 6.33... for 50.

Answer 3

What's the formula for mean absolute deviation?

Answer 4

Are you sure? The average should be 60 and each of the distances are: 10,0,10,15,5,10. I got those, at least.

Answer 5

\[\Large
\begin{array}{rcl}
\frac{1}{6}\sum_i^6x_i &=& \mu \\
\frac{1}{6}\sum_i^6|x_i-\mu| &=& 8.3\\
\end{array}
\]

Answer 6

Its basically the average of the absolute difference between the mean and each number.

Answer 7

\[
\sum_i^6x_i = 50+60+70+75+55 + x_6
\]

Answer 8

It's kinda tough isn't it. You need \(x_6\) to find the \(\mu\) and \(\mu\) to find \(x_6\)

Answer 9

Yes, It is 50; could you solve it algebraically?

Answer 10

50

Answer 11

I know it is 50; I just don't know how to solve it.

Answer 12

let x = # of cans collected on day 6

the mean is

M = (50+60+70+75+55+x)/6

M = (310+x)/6

Answer 13

I know the mean is (310 + x)/6 and I subtract that from each value, but because it is absolute value I'm confused

Answer 14

How do I know which difference is positive or negative?

Answer 15

Now subtract this mean from each data value and take the absolute value

| (310+x)/6 - 50 | = | (10+x)/6 |

| (310+x)/6 - 60 | = | (-50+x)/6 |

| (310+x)/6 - 70 | = | (-110+x)/6 |

| (310+x)/6 - 75 | = | (-140+x)/6 |

| (310+x)/6 - 55 | = | (-20+x)/6 |

| (310+x)/6 - x | = | (310-5x)/6 |

Answer 16

What is the absolute value of | (-20+x)/6 | ?

Answer 17

unfortunately there are two cases

| (-20+x)/6 | = (-20+x)/6 if x >= 20

| (-20+x)/6 | = -(-20+x)/6 if x < 20

Answer 18

So, if there are two cases, how do I determine which case to use?

Answer 19

the good news however is that

| (10+x)/6 | = (10+x)/6

since x > 0

Answer 20

the rest will have two cases, which may be a pain

so the best way to solve is to use a graphing calculator

Answer 21

I'm not allowed to use a graphing calculator, only a scientific.

Answer 22

then you'll have to try each case separately

Answer 23

or you can plug in each answer choice and you'll get the answer that way

Answer 24

I'm supposed to be able to solve a question similar to this in under 2 minutes. Which way would be faster?

Answer 25

i guess since your answer choices are at least 50 or over, you can conclude that

| (-50+x)/6 | = (-50+x)/6

and

| (-20+x)/6 | = (-20+x)/6

Answer 26

so that saves you a bit of work

Answer 27

@jim_thompson5910

Answer 28

what's up?

Answer 29

When I tried solving the problem, I got (500-4x)/6 = 8.3, but when I solve for x, I get 112.55 which is way off any of the choices.

Answer 30

how did you get that?

Answer 31

| (310+x)/6 - 50 | = | (10+x)/6 |
| (310+x)/6 - 60 | = | (-50+x)/6 |
| (310+x)/6 - 70 | = | (110-x)/6 | (because x > 50)
| (310+x)/6 - 75 | = | (-140+x)/6 | (because x>50)
| (310+x)/6 - 55 | = | (-20+x)/6 |
| (310+x)/6 - x | = | (310-5x)/6 |

Answer 32

Ignore the absolute value symbols.

Answer 33

Add them all up, then divide that by 6, set that equal to 8.3

[ (x+10)/6+(-50+x)/6+(110-x)/6+(140-x)/6+(-20+x)/6+(310-5x)/6 ] /6 = 8.3

(x+10)/6+(-50+x)/6+(110-x)/6+(140-x)/6+(-20+x)/6+(310-5x)/6 = 8.3*6

(x+10)/6+(-50+x)/6+(110-x)/6+(140-x)/6+(-20+x)/6+(310-5x)/6 = 49.8

I'll let you finish

Answer 34

@jim_thompson5910

Answer 35

so what do you get when you solve for x

Answer 36

112, I don't know where I messed up in simple addition.

Answer 37

(x+10)/6+(-50+x)/6+(110-x)/6+(140-x)/6+(-20+x)/6+(310-5x)/6 = 49.8

(x+10-50+x+110-x+140-x-20+x+310-5x)/6 = 49.8

(-4x + 500)/6 = 49.8

-4x + 500 = 49.8*6

-4x + 500 = 298.8

-4x = 298.8-500

-4x = -201.2

x = -201.2/(-4)

x = 50.3

Answer 38

Wait, didn't you already multiply by 6?

Answer 39

8.3 * 6 * 6 = 298.8

Answer 40

well for each term you're dividing by 6 (since the mean has you do that)

when you find the MAD, you are dividing the total sum by 6

Answer 41

so that explains why there are 2 instances where you're multiplying by 6

Answer 42

You kind of lost me there.

Answer 43

you saw how we got

| (310+x)/6 - 50 | = | (10+x)/6 |

| (310+x)/6 - 60 | = | (-50+x)/6 |

| (310+x)/6 - 70 | = | (-110+x)/6 |

| (310+x)/6 - 75 | = | (-140+x)/6 |

| (310+x)/6 - 55 | = | (-20+x)/6 |

| (310+x)/6 - x | = | (310-5x)/6 |

right?

Answer 44

Wait, nevermind, I think I get it.

Answer 45

oh ok great

Answer 46

Each difference was divided by six and you divide the sum of the differences by 6, so you multiply by 36 on both sides of the equation to get rid of the denominators.

Answer 47

exactly

Answer 48

Thanks!

Answer 49

yw