Answer:
![\begin{array}{ccccccccccccccc}{1} & {3} & {4} & {12} & {15} & {18}& {18 } & {26} & {29} & {30} & {32} & {57} & {58} & {61} \\ \end{array}](https://tex.z-dn.net/?f=%5Cbegin%7Barray%7D%7Bccccccccccccccc%7D%7B1%7D%20%26%20%7B3%7D%20%26%20%7B4%7D%20%26%20%7B12%7D%20%26%20%7B15%7D%20%26%20%7B18%7D%26%20%7B18%20%7D%20%26%20%7B26%7D%20%26%20%7B29%7D%20%26%20%7B30%7D%20%26%20%7B32%7D%20%26%20%7B57%7D%20%26%20%7B58%7D%20%26%20%7B61%7D%20%5C%5C%20%5Cend%7Barray%7D)
Step-by-step explanation:
Given
![\begin{array}{ccccccccccccccc}{1} & {[ \ ]} & {4} & {[ \ ] } & {15} & {18}& {[ \ ] } & {[ \ ]} & {29} & {30} & {32} & {[ \ ]} & {58} & {[ \ ]} \\ \end{array}](https://tex.z-dn.net/?f=%5Cbegin%7Barray%7D%7Bccccccccccccccc%7D%7B1%7D%20%26%20%7B%5B%20%5C%20%5D%7D%20%26%20%7B4%7D%20%26%20%7B%5B%20%5C%20%5D%20%7D%20%26%20%7B15%7D%20%26%20%7B18%7D%26%20%7B%5B%20%5C%20%5D%20%7D%20%26%20%7B%5B%20%5C%20%5D%7D%20%26%20%7B29%7D%20%26%20%7B30%7D%20%26%20%7B32%7D%20%26%20%7B%5B%20%5C%20%5D%7D%20%26%20%7B58%7D%20%26%20%7B%5B%20%5C%20%5D%7D%20%5C%5C%20%5Cend%7Barray%7D)
Required
Fill in the box
From the question, the range is:
![Range = 60](https://tex.z-dn.net/?f=Range%20%3D%2060)
Range is calculated as:
![Range = Highest - Least](https://tex.z-dn.net/?f=Range%20%3D%20%20Highest%20-%20Least)
From the box, we have:
![Least = 1](https://tex.z-dn.net/?f=Least%20%3D%201)
So:
![60 = Highest - 1](https://tex.z-dn.net/?f=60%20%3D%20Highest%20%20-%201)
![Highest = 60 +1](https://tex.z-dn.net/?f=Highest%20%3D%2060%20%2B1)
![Highest = 61](https://tex.z-dn.net/?f=Highest%20%3D%2061)
The box, becomes:
![\begin{array}{ccccccccccccccc}{1} & {[ \ ]} & {4} & {[ \ ] } & {15} & {18}& {[ \ ] } & {[ \ ]} & {29} & {30} & {32} & {[ \ ]} & {58} & {61} \\ \end{array}](https://tex.z-dn.net/?f=%5Cbegin%7Barray%7D%7Bccccccccccccccc%7D%7B1%7D%20%26%20%7B%5B%20%5C%20%5D%7D%20%26%20%7B4%7D%20%26%20%7B%5B%20%5C%20%5D%20%7D%20%26%20%7B15%7D%20%26%20%7B18%7D%26%20%7B%5B%20%5C%20%5D%20%7D%20%26%20%7B%5B%20%5C%20%5D%7D%20%26%20%7B29%7D%20%26%20%7B30%7D%20%26%20%7B32%7D%20%26%20%7B%5B%20%5C%20%5D%7D%20%26%20%7B58%7D%20%26%20%7B61%7D%20%5C%5C%20%5Cend%7Barray%7D)
From the question:
--- interquartile range
This is calculated as:
![IQR = Q_3 - Q_1](https://tex.z-dn.net/?f=IQR%20%3D%20Q_3%20-%20Q_1)
is the median of the upper half while
is the median of the lower half.
So, we need to split the given boxes into two equal halves (7 each)
<u>Lower half:</u>
![\begin{array}{ccccccc}{1} & {[ \ ]} & {4} & {[ \ ] } & {15} & {18}& {[ \ ] } \\ \end{array}](https://tex.z-dn.net/?f=%5Cbegin%7Barray%7D%7Bccccccc%7D%7B1%7D%20%26%20%7B%5B%20%5C%20%5D%7D%20%26%20%7B4%7D%20%26%20%7B%5B%20%5C%20%5D%20%7D%20%26%20%7B15%7D%20%26%20%7B18%7D%26%20%7B%5B%20%5C%20%5D%20%7D%20%5C%5C%20%5Cend%7Barray%7D)
<u>Upper half</u>
<u></u>
<u></u>
The quartile is calculated by calculating the median for each of the above halves is calculated as:
![Median = \frac{N + 1}{2}th](https://tex.z-dn.net/?f=Median%20%3D%20%5Cfrac%7BN%20%2B%201%7D%7B2%7Dth)
Where N = 7
So, we have:
![Median = \frac{7 + 1}{2}th = \frac{8}{2}th = 4th](https://tex.z-dn.net/?f=Median%20%3D%20%5Cfrac%7B7%20%2B%201%7D%7B2%7Dth%20%3D%20%5Cfrac%7B8%7D%7B2%7Dth%20%3D%204th)
So,
= 4th item of the upper halves
= 4th item of the lower halves
From the upper halves
<u></u>
<u></u>
<u></u>
We have:
![Q_3 = 32](https://tex.z-dn.net/?f=Q_3%20%3D%2032)
can not be determined from the lower halves because the 4th item is missing.
So, we make use of:
![IQR = Q_3 - Q_1](https://tex.z-dn.net/?f=IQR%20%3D%20Q_3%20-%20Q_1)
Where
and ![IQR = 20](https://tex.z-dn.net/?f=IQR%20%3D%2020)
So:
![20 = 32 - Q_1](https://tex.z-dn.net/?f=20%20%3D%2032%20-%20Q_1)
![Q_1 = 32 - 20](https://tex.z-dn.net/?f=Q_1%20%3D%2032%20-%2020)
![Q_1 = 12](https://tex.z-dn.net/?f=Q_1%20%3D%2012)
So, the lower half becomes:
<u>Lower half:</u>
![\begin{array}{ccccccc}{1} & {[ \ ]} & {4} & {12 } & {15} & {18}& {[ \ ] } \\ \end{array}](https://tex.z-dn.net/?f=%5Cbegin%7Barray%7D%7Bccccccc%7D%7B1%7D%20%26%20%7B%5B%20%5C%20%5D%7D%20%26%20%7B4%7D%20%26%20%7B12%20%7D%20%26%20%7B15%7D%20%26%20%7B18%7D%26%20%7B%5B%20%5C%20%5D%20%7D%20%5C%5C%20%5Cend%7Barray%7D)
From this, the updated values of the box is:
![\begin{array}{ccccccccccccccc}{1} & {[ \ ]} & {4} & {12} & {15} & {18}& {[ \ ] } & {[ \ ]} & {29} & {30} & {32} & {[ \ ]} & {58} & {61} \\ \end{array}](https://tex.z-dn.net/?f=%5Cbegin%7Barray%7D%7Bccccccccccccccc%7D%7B1%7D%20%26%20%7B%5B%20%5C%20%5D%7D%20%26%20%7B4%7D%20%26%20%7B12%7D%20%26%20%7B15%7D%20%26%20%7B18%7D%26%20%7B%5B%20%5C%20%5D%20%7D%20%26%20%7B%5B%20%5C%20%5D%7D%20%26%20%7B29%7D%20%26%20%7B30%7D%20%26%20%7B32%7D%20%26%20%7B%5B%20%5C%20%5D%7D%20%26%20%7B58%7D%20%26%20%7B61%7D%20%5C%5C%20%5Cend%7Barray%7D)
From the question, the median is:
and ![N = 14](https://tex.z-dn.net/?f=N%20%3D%2014)
To calculate the median, we make use of:
![Median = \frac{N + 1}{2}th](https://tex.z-dn.net/?f=Median%20%3D%20%5Cfrac%7BN%20%2B%201%7D%7B2%7Dth)
![Median = \frac{14 + 1}{2}th](https://tex.z-dn.net/?f=Median%20%3D%20%5Cfrac%7B14%20%2B%201%7D%7B2%7Dth)
![Median = \frac{15}{2}th](https://tex.z-dn.net/?f=Median%20%3D%20%5Cfrac%7B15%7D%7B2%7Dth)
![Median = 7.5th](https://tex.z-dn.net/?f=Median%20%3D%207.5th)
This means that, the median is the average of the 7th and 8th items.
The 7th and 8th items are blanks.
However, from the question; the mode is:
![Mode = 18](https://tex.z-dn.net/?f=Mode%20%3D%2018)
Since the values of the box are in increasing order and the average of 18 and 18 do not equal 22 (i.e. the median), then the 7th item is:
![7th = 18](https://tex.z-dn.net/?f=7th%20%3D%2018)
The 8th item is calculated as thus:
![Median = \frac{1}{2}(7th + 8th)](https://tex.z-dn.net/?f=Median%20%3D%20%5Cfrac%7B1%7D%7B2%7D%287th%20%2B%208th%29)
![22= \frac{1}{2}(18 + 8th)](https://tex.z-dn.net/?f=22%3D%20%5Cfrac%7B1%7D%7B2%7D%2818%20%2B%208th%29)
Multiply through by 2
![44 = 18 + 8th](https://tex.z-dn.net/?f=44%20%3D%2018%20%2B%208th)
![8th = 44 - 18](https://tex.z-dn.net/?f=8th%20%3D%2044%20-%2018)
![8th = 26](https://tex.z-dn.net/?f=8th%20%3D%2026)
The updated values of the box is:
![\begin{array}{ccccccccccccccc}{1} & {[ \ ]} & {4} & {12} & {15} & {18}& {18 } & {26} & {29} & {30} & {32} & {[ \ ]} & {58} & {61} \\ \end{array}](https://tex.z-dn.net/?f=%5Cbegin%7Barray%7D%7Bccccccccccccccc%7D%7B1%7D%20%26%20%7B%5B%20%5C%20%5D%7D%20%26%20%7B4%7D%20%26%20%7B12%7D%20%26%20%7B15%7D%20%26%20%7B18%7D%26%20%7B18%20%7D%20%26%20%7B26%7D%20%26%20%7B29%7D%20%26%20%7B30%7D%20%26%20%7B32%7D%20%26%20%7B%5B%20%5C%20%5D%7D%20%26%20%7B58%7D%20%26%20%7B61%7D%20%5C%5C%20%5Cend%7Barray%7D)
From the question.
![Mean = 26](https://tex.z-dn.net/?f=Mean%20%3D%2026)
Mean is calculated as:
![Mean = \frac{\sum x}{n}](https://tex.z-dn.net/?f=Mean%20%3D%20%5Cfrac%7B%5Csum%20x%7D%7Bn%7D)
So, we have:
![26= \frac{1 + 2nd + 4 + 12 + 15 + 18 + 18 + 26 + 29 + 30 + 32 + 12th + 58 + 61}{14}](https://tex.z-dn.net/?f=26%3D%20%5Cfrac%7B1%20%2B%202nd%20%2B%204%20%2B%2012%20%2B%2015%20%2B%2018%20%2B%2018%20%2B%2026%20%2B%2029%20%2B%2030%20%2B%2032%20%2B%2012th%20%2B%2058%20%2B%2061%7D%7B14%7D)
Collect like terms
![26= \frac{ 2nd + 12th+1 + 4 + 12 + 15 + 18 + 18 + 26 + 29 + 30 + 32 + 58 + 61}{14}](https://tex.z-dn.net/?f=26%3D%20%5Cfrac%7B%202nd%20%2B%2012th%2B1%20%2B%204%20%2B%2012%20%2B%2015%20%2B%2018%20%2B%2018%20%2B%2026%20%2B%2029%20%2B%2030%20%2B%2032%20%2B%2058%20%2B%2061%7D%7B14%7D)
![26= \frac{ 2nd + 12th+304}{14}](https://tex.z-dn.net/?f=26%3D%20%5Cfrac%7B%202nd%20%2B%2012th%2B304%7D%7B14%7D)
Multiply through by 14
![14 * 26= 2nd + 12th+304](https://tex.z-dn.net/?f=14%20%2A%2026%3D%202nd%20%2B%2012th%2B304)
![364= 2nd + 12th+304](https://tex.z-dn.net/?f=364%3D%202nd%20%2B%2012th%2B304)
This gives:
![2nd + 12th = 364 - 304](https://tex.z-dn.net/?f=2nd%20%2B%2012th%20%3D%20364%20-%20304)
![2nd + 12th = 60](https://tex.z-dn.net/?f=2nd%20%2B%2012th%20%3D%2060)
From the updated box,
![\begin{array}{ccccccccccccccc}{1} & {[ \ ]} & {4} & {12} & {15} & {18}& {18 } & {26} & {29} & {30} & {32} & {[ \ ]} & {58} & {61} \\ \end{array}](https://tex.z-dn.net/?f=%5Cbegin%7Barray%7D%7Bccccccccccccccc%7D%7B1%7D%20%26%20%7B%5B%20%5C%20%5D%7D%20%26%20%7B4%7D%20%26%20%7B12%7D%20%26%20%7B15%7D%20%26%20%7B18%7D%26%20%7B18%20%7D%20%26%20%7B26%7D%20%26%20%7B29%7D%20%26%20%7B30%7D%20%26%20%7B32%7D%20%26%20%7B%5B%20%5C%20%5D%7D%20%26%20%7B58%7D%20%26%20%7B61%7D%20%5C%5C%20%5Cend%7Barray%7D)
We know that:
<em>The 2nd value can only be either 2 or 3</em>
<em>The 12th value can take any of the range 33 to 57</em>
Of these values, the only possible values of 2nd and 12th that give a sum of 60 are:
![2nd = 3](https://tex.z-dn.net/?f=2nd%20%3D%203)
![12th = 57](https://tex.z-dn.net/?f=12th%20%3D%2057)
i.e.
![2nd + 12th = 60](https://tex.z-dn.net/?f=2nd%20%2B%2012th%20%3D%2060)
![3 + 57 = 60](https://tex.z-dn.net/?f=3%20%2B%2057%20%3D%2060)
So, the complete box is:
![\begin{array}{ccccccccccccccc}{1} & {3} & {4} & {12} & {15} & {18}& {18 } & {26} & {29} & {30} & {32} & {57} & {58} & {61} \\ \end{array}](https://tex.z-dn.net/?f=%5Cbegin%7Barray%7D%7Bccccccccccccccc%7D%7B1%7D%20%26%20%7B3%7D%20%26%20%7B4%7D%20%26%20%7B12%7D%20%26%20%7B15%7D%20%26%20%7B18%7D%26%20%7B18%20%7D%20%26%20%7B26%7D%20%26%20%7B29%7D%20%26%20%7B30%7D%20%26%20%7B32%7D%20%26%20%7B57%7D%20%26%20%7B58%7D%20%26%20%7B61%7D%20%5C%5C%20%5Cend%7Barray%7D)