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Each of 100 balls has an integer value from 1 to 8, inclusiv
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16 Sep 2017, 11:08
16 Sep 2017, 11:08
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Question Stats:
31% (02:34) correct
68% (03:02) wrong based on 249 sessions
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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
Re: Each of 100 balls has an integer value from 1 to 8, inclusiv
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17 Nov 2017, 02:15
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
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
General Discussion
Re: Each of 100 balls has an integer value from 1 to 8, inclusiv
[#permalink]
22 Jun 2021, 08:34
22 Jun 2021, 08:34
6
The “interquartile range” of a group of 100 integers is found by
splitting 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. Order the
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.
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 Q3
is the average of 5 and 6, or 5.5. Count carefully—if you are off by even just
one either way, you’ll get a different number for the third quartile. Balls #25
and #26 both have a 3 on them. So the first quartile Q1
is the average of 3 and
3, namely 3.
Finally, Q3 – Q1 = 5.5 – 3 = 2.5.
splitting 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. Order the
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.
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 Q3
is the average of 5 and 6, or 5.5. Count carefully—if you are off by even just
one either way, you’ll get a different number for the third quartile. Balls #25
and #26 both have a 3 on them. So the first quartile Q1
is the average of 3 and
3, namely 3.
Finally, Q3 – Q1 = 5.5 – 3 = 2.5.
