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02 Feb 2012, 04:34
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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:
b. What is the greatest possible number of seniors whose theses were within six pages of the median length ?
Answer:
17
Attachment:
2.png [ 79.91 KiB | Viewed 29729 times ]
02 Feb 2012, 04:57
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abhi47
Each senior in a college course wrote a thesis. The lengths in pages of those seniors's theses are summarized in the graph.
a) What is the least possible number of seniors whose theses were within six pages of the median length ?
B) What is the greatest possible number of seniors whose theses were within six pages of the median length ?
The graph is attached as a jpg file .
Could someone provide a simpler solution to this problem ? The actual solution provided is quite confusing.
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.
07 Feb 2012, 03:31
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abhi47
Each senior in a college course wrote a thesis. The lengths in pages of those seniors's theses are summarized in the graph.
a) What is the least possible number of seniors whose theses were within six pages of the median length ?
B) What is the greatest possible number of seniors whose theses were within six pages of the median length ?
a) What is the least possible number of seniors whose theses were within six pages of the median length ?
B) What is the greatest possible number of seniors whose theses were within six pages of the median length ?
Attachment:
WpS6j.png
Could someone provide a simpler solution to this problem ? The actual solution provided is quite confusing.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, [highlight]20, 20[/highlight], 30, 30, 30, 30, 30, 30, 30, 40, 40
or
List 2: 5, 10, 12, 12, 13, 20, 20, 20, 20, [highlight]29, 29[/highlight], 39, 39, 39, 39, 39, 39, 39, 40, 40
or
List 3: 5, 10, 12, 12, 13, 20, 20, 20, 20, [highlight]27, 29[/highlight], 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, [highlight]20, 29[/highlight], 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.
