Each senior in a college course wrote a thesis. The lengths

Question

https://gmatclub.com/forum/download/file.php?id=20343 Each senior in a college course wrote a thesis. The lengths, in pages, of those seniors' theses are summarized in the graph above. a. What is the least possible number of seniors whose theses were within six pages of the median length ? Answer: 2 b. What is the greatest possible number of seniors whose theses were within six pages of the median length ? Answer: 17 2.png

Bunuel · Accepted Answer

There are total of 20 students.
1st group: 1 student wrote between 0-9 pages;
2nd group: 4 students wrote between 10-19 pages;
3rd group: 6 students wrote between 20-29 pages;
4th group: 7 students wrote between 30-39 pages;
5th group: 2 students wrote between 40-49 pages.

As there are 20 students, then the median length would be the average of 10th and 11th students writings. Both, 10th and 11th students are in the third group, so both these student wrote between 20 and 29 pages (so the median will be between 20 and 29).

A. We want to minimize the # of writings which fall in the range median-6 and median+6 pages. We want to spread the # of writings as far as possible. Make, all but 10th and 11th students from 3rd group to write 20 pages (min possible for this group), all student from 4th group to write 39 pages (max possible for this group) and 10th and 11th students to write 29 pages each, then median=(29+29)/2=29 and only these 2 student fall in the range 29-6=23 and 29+6=35 pages. Note that less than 2 is not possible as even if we put as far apart as possible 10th and 11th students' writings, 20 pages for 10th and 29th pages for 11th, then the median=(20+29)/2=24.5 and still these 2 student (at least) fall in the range median-6=18.5 and median+6=30.5 pages.

Answer: 2.

B. We want to maximize the # of writings which fall in the range median-6 and median+6 pages. We want to group # of writings as close as possible to the median, which still will be between 20 and 29. Now, make ALL 4 students from the second group to write 19 pages (max possible for this group) and ALL 7 students from the fourth group to write 30 pages (min possible for this group). Also make 10th and 11th students form the third group to write 24 and 25 pages, so that the median will be (24+25)/2=24.5. In this cases all these writings will fall in the range 24.5-6=18.5 and 24.5+6=30.5, so total of 4+7+6=17 writings.

Answer: 17.

KarishmaB · Answer

The graph gives us the following information:
No of pages : No of people
0-9 : 1
10 - 19 : 4
20 - 29 : 6
30 - 39 : 7
40 - 49 : 2

There are a total of 20 people (1+4+6+7+2)
The median number of pages will be the mean of the 10th and the 11th terms. 
The 10th and 11th seniors lie in the 20-29 pages interval. So the median will lie in the 20 - 29 interval.

The list of the number of pages could look something like this:
List 1: 5, 10, 10, 10, 10, 20, 20, 20, 20,  20, 20 , 30, 30, 30, 30, 30, 30, 30, 40, 40
or
List 2: 5, 10, 12, 12, 13, 20, 20, 20, 20,  29, 29 , 39, 39, 39, 39, 39, 39, 39, 40, 40
or 
List 3: 5, 10, 12, 12, 13, 20, 20, 20, 20,  27, 29 , 36, 36, 36, 36, 36, 36, 39, 40, 40
or many many other ways

The highlighted terms are the 10th and the 11th terms.

a) What is the least possible number of seniors whose theses were within six pages of the median length ?

We need to move people away from the 10th and the 11th terms such that they are not within 6 pages of median.
Median can be 20 or 29 or anything in between (e.g. (28+29)/2 = 28.5 or (20+29)/2 = 24.5 etc)
In list 1, median is 20. In list 2, median is 29 (there are only two people within a distance of 6 from 29). 
Is it possible to move ALL people 6 pages away from median? No. We have 4 more people in the 20-29 range and even if they all are at 20, the 10th person will be at 27 and 11th will be at 29. So the median will be 28. (see list 3)

So the 10th and the 11th person will always be within 6 pages of the median.

B) What is the greatest possible number of seniors whose theses were within six pages of the median length ?
We try to bring the people from all groups as close as possible to the median. 
Only people from 10-19, 20-29 and 30-39 can possibly be 6 pages away. Let's bring the median right in the middle so that 10-19 and 30-39 can come within 6 pages of the median.

List: 5, 19, 19, 19, 19, 20, 20, 20, 20,  20, 29 , 30, 30, 30, 30, 30, 30, 30, 40, 40
The median here will be (20+29)/2 = 24.5
Therefore we can have 4+6+7 = 17 seniors lie within 6 pages of the median.

sdas · Answer

Bunuel, I do not understand the explanation. I got median between 10-11th students median range of 20-29 pages. I lost you from there....request an explanation if possible to help me understand

Bunuel · Answer

A. In order to to minimize the # of writings which fall in the range median-6 and median+6 pages we should move as many writing as far as possible from the median. As the median is between 20-29 pages long (the third group) then move 4 students writings from this group to its lowest limit (20) and move all 7 students writings from the 4th group to its highest limit (39). Next, if 2 other students from the third group wrote 29 pages each, then the median would be 29, and only those two students will fall in the range from 29-6=23 and 29+6=35. Total: 2.

B. In order to to maximize the # of writings which fall in the range median-6 and median+6 pages we should move as many writing as close as possible from the median. The same here: move all 4 students writings from the 2nd group to its highest limit (19), move all 7 students writings from the 4th group to its lowest (30). Next, make the median so that median-6<19 (in order 4 students from the 2nd group to fall in the range) and median+6>30 (in order 7 students from the 4th group to fall in the range), which basically means that the median should be (19+30)/2=24.5. So, 10th and 11th students form the third group should write 24 and 25 pages (obviouslyall other students from this group will also be in the range). Total: 4+6+7=17.

Hope it's clear.

sdas · Answer

Sorry Bunuel, I still cannot get there. Your explanation is still not clear to me. I am very weak in statistics, if you could please break it down a bit more, perhaps I will get there. Thanks in advance

Bunuel · Answer

Can you please specify what part didn't you get? Check similar questions to practice: 5-peices-of-wood-have-an-average-length-of-124-inches-and-123513.html in-a-class-of-30-students-2-students-did-not-borrow-any-101980.html set-r-contains-five-numbers-that-have-an-average-value-of-102087.html three-boxes-of-supplies-have-an-average-arithmetic-mean-105819.html the-average-arithmetic-mean-of-the-5-positive-integers-k-107059.html if-the-average-of-5-positive-integers-is-40-and-the-127038.html Hope it helps.

PiyushK · Answer

I tried to calculate median using cumulative frequency and got 27.333 + - 6 => 21.3 and 33.3.

Using this range i got 0 and 13 answer.

Attachment:

median.jpg [ 54.4 KiB | Viewed 29672 times ]

Is this method applicable to this question ? or I am using a wrong method please help to verify ?

Thanks.

KarishmaB · Answer

I am not sure what you have done here and why. How did you get a value for median? The median can take various different values. Also median is not a range or two points (21.3 and 33.3 that you obtained). Median of a data set is a single value. Are you somehow using Standard Deviation here?

PiyushK · Answer

Hi Karishma,

I found this formula to calculate median for data defined in intervals.
Refer following link scroll down to median's formula.
https://mba-lectures.com/statistics/desc ... -data.html

I am just doing plus minus 6 pages to the median page count.
In this range I can keep minimum 0 seniors and maximum all 13 seniors.

Regards
PiyushK

KarishmaB · Answer

The two concepts are different. In simple terms, the method you used is used to "best guess" the value of the median. Say, we only have the data of the graph and we need to guess the value of the median, then we can use that method. Actually, the value of the median might be much different. This question does not make you guess the value of the median. It says that the value can vary and asks for the case where you will get least/most members. It asks you how you should allot the pages so that you get the least/most value.

smallapple · Answer

Hi Bunuel,
I know the median length is the average length of the 10th and 11th theses, but in the group where all of the 6 theses have 20-29 pages each, the 10th and the 11th theses are in the middle in terms of their lengths, therefore, given both 10th and 11th theses have 29 pages, the 8th and 9th theses in the 20-29 group have 22 pages each , and all of the 7 theses in the 30-39 group have 36 pages, I think the least number of the seniors writing theses satisfying the length within the range is 4, rather than 2, because the length of the 12th and 13th theses in the 20-29 group cannot be shorter than the 10th and 11th theses in the same group. Am I right? Thank you.

Bunuel · Answer

No, you are not right. Consider example below:

{0} {10, 10, 10, 10} {20, 20, 20, 20,   29, 29  } {39, 39, 39, 39, 39, 39, 39} {40, 40}

1st group: 1 student wrote between 0-9 pages;
2nd group: 4 students wrote between 10-19 pages;
3rd group: 6 students wrote between 20-29 pages;
4th group: 7 students wrote between 30-39 pages;
5th group: 2 students wrote between 40-49 pages.

Hope it helps.

futurephilantropist · Answer

Hello,

Can somebody explain how median of 10 and 11 fall into 6 students' pages? if we list students as following: 
1 student - 0-9 pages
2 students - 40-49 pages
7 students - 30-39 pages
4 students - 10-19 pages
6 students - 20-29 pages
then, median of 10-11 students won't correspond to 20-29 pages.

Bunuel · Answer

The median of a set with even number of elements is the average of two middle numbers  when arranged in  ascending  or  descending  order .

There are total of 20 students.
1st group: 1 student wrote between 0-9 pages, say 1 page each;
2nd group: 4 students wrote between 10-19 pages, say 10 pages each;
 3rd group: 6 students wrote between 20-29 pages, say 20 pages each; 
4th group: 7 students wrote between 30-39 pages, say 30 pages each;
5th group: 2 students wrote between 40-49 pages, say 40 pages each.

{ 1 ,  10, 10, 10, 10,   20, 20, 20, 20,  20, 20,    30, 30, 30, 30, 30, 30, 30,   40, 40 }.

As you can see, the median length would be the average of 10th and 11th students writings. Both, 10th and 11th students are in the third group, so both these student wrote between 20 and 29 pages (so the median will be between 20 and 29).

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Each senior in a college course wrote a thesis. The lengths

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