Problem Solving

I understand what the problem is asking and I understand how they got the solution, but I'm confused as to why the solution is correct.

The problem is asking for the average value of two average values, so why divide again by all 12 packages when the individual average values were already divided by 8 and 4 respectively? It seems like we are summing up the values and then dividing by 12 twice.

I even tried making up a simplified example to see what would happen if all the individual values were known:

Average of 1, 2, 3, 4 = (1 + 2+ 3 + 4)/4 = 5/2.
Average of 1,2 = (1 + 2)/2 = 3/2.

Average of all of the above values = (1 + 2 + 3 + 4 + 1 + 2)/6 = 13/6

But if I were to use the OG solution on this simplified example, it would be
(5/2) + (3/2) / 6 = (8/2)/6 = 8/12 = 2/3

2/3 is not equal to 13/6.

I'm missing some logic that the book just isn't detailing. I really want to understand this problem. Can anyone help?

bekkilyn wrote:I understand what the problem is asking and I understand how they got the solution, but I'm confused as to why the solution is correct.

The problem is asking for the average value of two average values, so why divide again by all 12 packages when the individual average values were already divided by 8 and 4 respectively? It seems like we are summing up the values and then dividing by 12 twice.

I even tried making up a simplified example to see what would happen if all the individual values were known:

Average of 1, 2, 3, 4 = (1 + 2+ 3 + 4)/4 = 5/2.
Average of 1,2 = (1 + 2)/2 = 3/2.

Average of all of the above values = (1 + 2 + 3 + 4 + 1 + 2)/6 = 13/6

But if I were to use the OG solution on this simplified example, it would be
(5/2) + (3/2) / 6 = (8/2)/6 = 8/12 = 2/3

2/3 is not equal to 13/6.

I'm missing some logic that the book just isn't detailing. I really want to understand this problem. Can anyone help?

I think the question is SPECIFICALLY asking you to take the average of the average of two sets. So, you just can't take the average of the two set combined as one set

Hi Bekkilyn,
This is weighted average question since the number of packages have different values 8 and 4(with their own mean weights)

So
(8*mean weight of the 1st 8 packages + 4*mean weight of 2nd 4 packages) divided by total packages(12) gives you the combined mean weight of 12 packages step1

Its not the same as adding the individual mean wieghts and dividing by 2


For example lets say the number of packages was the same(with different mean weights) then the weighted average and average would be the same.

Lest say 4 packages have a mean wieght of 2 and 4 other packages have a mean weight of 3

So doing4*2 + 4*3 / 8 = 20/8 = 2.5 is the same as adding the 2 means and dividing by 2 i.e. 2+3/2 = 2.5
Since the number of packages are different(8 and 4) with their own respective mean weights in the OG PROBLEM you would have to do like whats mentioned in step 1


Hope this helps!

*Weighted* mean vs. *arithmetic* mean...yes! I'd completely forgotten about weighted mean, but after reading these replies, I went and pulled out my trusty statistics text and there it was...weighted mean....plain as an uncracked egg.

This problem solution makes much more sense now! Thank you!