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# Each of 100 balls has an integer value from 1 to 8, inclusiv

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Each of 100 balls has an integer value from 1 to 8, inclusiv [#permalink]  16 Sep 2017, 11:08
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Each of 100 balls has an integer value from 1 to 8, inclusive, painted on the surface. The number $$Nx$$ of balls representing integer x is defined by the formula Nx = 18 (x – 4)^2.

What is the interquartile range of the 100 integers?

(A) 1.5

(B) 2.0

(C) 2.5

(D) 3.0

(E) 3.5
[Reveal] Spoiler: OA

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Re: Each of 100 balls has an integer value from 1 to 8, inclusiv [#permalink]  17 Nov 2017, 02:15
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Solution: Before figuring out how many balls you have of each integer value, consider what the question is asking: the “interquartile range” of a group of 100 integers. To find this range, split the 100 integers into two groups, a lower 50 and an upper 50. Then find the median of each of those groups. The median of the lower group is the first quartile (Q1), while the median of the upper group is the third quartile (Q3). Finally, Q3-Q1 is the interquartile range.
The median of a group of 50 integers is the average (arithmetic mean) of the 25th and the 26th integers when ordered from smallest to largest. Out of the ordered list of 100 integers from smallest to largest, then, find #25 and #26 and average them to get the first quartile. Likewise, find #75 and #76 and average them to get the third quartile. Then perform the subtraction.
Each ball has an integer value painted on the side — either 1, 2, 3, 4, 5, 6, 7, or 8. Figure out how many balls there are for each integer by applying the given formula, starting with the lowest integer in the list (1) and going up from there.
Number of balls labeled number 1 = 18 - (1 - 4)2 = 18 - (-3)2 = 18 - 9 = 9 balls. These represent balls #1 through #9.
Number of balls labeled number 2 = 18 - (2 - 4)2 = 18 - (-2)2 = 18 - 4 = 14 balls, representing balls #10 through #23.
Be careful when counting; the 14th ball is #23, not #24, because #10 is the first, #11 is the second, and so on.
Number of balls labeled number 3 = 18 - (3 - 4)2 = 18 - (-1)2 = 18 - 1 = 17 balls, representing #24 through #40.
At this point, you can tell that balls #25 and #26 both have a 3 on them. So the first quartile Q1 is the average of 3 and 3, namely 3. Now keep going!
Number of balls labeled number 4 = 18 - (4 - 4)2 = 18 - (0)2 = 18 balls, representing balls #41 through #58.
Number of balls labeled number 5 = 18 - (5 - 4)2 = 18 - (1)2 = 18 - 1 = 17 balls, representing #59 through #75.
You can stop here. Ball #75 has a 5 on it (in fact, the last 5), while ball #76 must have a 6 on it (since 6 is the next integer in the list). Thus, the third quartile Q1 is the average of 5 and 6, or 5.5. Notice that you have to count carefully — if you are off by even just one either way, you’ll get a different number for the third quartile.
Finally, Q3-Q1 = 5.5 - 3 = 2.5, the interquartile range of this list of integers
Re: Each of 100 balls has an integer value from 1 to 8, inclusiv   [#permalink] 17 Nov 2017, 02:15
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