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The five-number summary of a dataset is given.
Lowest score = 1
Lowest quartile (`Q_1`) = 4
Median (`Q_2`) = 7
Upper quartile (`Q_3`) = 10
Highest score = 20
Is 20 an outlier? Justify your answer with calculations. (2 marks)
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`text(20 is an outlier – See Worked Solution)`
`IQR = Q_3 – Q_1 = 10-4=6`
| `text(Upper Fence)` | `= Q_3 + 1.5 xx IQR` |
| `=10 + 1.5 xx 6` | |
| `=19` |
`text{S}text{ince 20 > 19, 20 is an outlier.}`
The heights, in centimetres, of 10 players on a basketball team are shown.
170, 180, 185, 188, 192, 193, 193, 194, 196, 202
Is the height of the shortest player on the team considered an outlier? Justify your answer with calculations. (3 marks)
`text(See Worked Solutions)`
`Q_1 = 185, \ Q_3 = 194`Mean mark 51%.
COMMENT: The last statement must be made to achieve full marks here!
`IQR = 194 – 185 = 9`
`text(Shortest player = 170)`
`text(Outlier height:)`
| `Q_1 – 1.5 xx IQR ` | `= 185 – 1.5 xx 9` |
| `= 171.5` |
`:.\ text(S)text(ince 170 < 171.5, 170 is an outlier.)`
A cumulative frequency table for a data set is shown.
\begin{array} {|c|c|}
\hline
\ \ \ \ \ \ \ \textit{Score}\ \ \ \ \ \ \ & \ \ \ \ \ \textit{Cumulative}\ \ \ \ \ \\ & \textit{frequency} \\
\hline
\rule{0pt}{2.5ex} \text{1} \rule[-1ex]{0pt}{0pt} & 5 \\
\hline
\rule{0pt}{2.5ex} \text{2} \rule[-1ex]{0pt}{0pt} & 9 \\
\hline
\rule{0pt}{2.5ex} \text{3} \rule[-1ex]{0pt}{0pt} & 16 \\
\hline
\rule{0pt}{2.5ex} \text{4} \rule[-1ex]{0pt}{0pt} & 20 \\
\hline
\rule{0pt}{2.5ex} \text{5} \rule[-1ex]{0pt}{0pt} & 34 \\
\hline
\rule{0pt}{2.5ex} \text{6} \rule[-1ex]{0pt}{0pt} & 42 \\
\hline
\end{array}
What is the interquartile range of this data set? (2 marks)
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`2`
`text(42 data points ⇒ median) = text(21st + 22nd)/2`♦♦ Mean mark 27%.
| `text(Q)_1` | `= 11text(th data point) = 3` |
| `text(Q)_3` | `= 32text(nd data point) = 5` |
| `:.\ text(IQR)` | `= 5 – 3` |
| `= 2` |
A set of data has a lower quartile (`Q_L`) of 10 and an upper quartile (`Q_U`) of 16.
What is the maximum possible range for this set of data if there are no outliers? (2 marks)
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`24`
`IQR = 16 – 10 = 6`
`text(If no outliers,)`
| `text(Upper limit)` | `= Q_U + 1.5 xx IQR` |
| `= 16 + 1.5 xx 6` | |
| `= 25` |
| `text(Lower limit)` | `= Q_L – 1.5 xx IQR` |
| `= 10 – 1.5 xx 6` | |
| `= 1` |
| `:.\ text(Maximum range)` | `= 25 – 1` |
| `= 24` |
In a small business, the seven employees earn the following wages per week:
\(\$300, \ \$490, \ \$520, \ \$590, \ \$660, \ \$680, \ \$970\)
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What effect will this have on the standard deviation? (1 mark)
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i. \(\text{See Worked Solutions.} \)
ii. \(\text{The standard deviation will remain the same.}\)
i. \(300, 490, 520, 590, 660, 680, 970\)
| \(\text{Median}\) | \(= 590\) |
| \(Q_1\) | \(= 490\) |
| \(Q_3\) | \(= 680\) |
| \(IQR\) | \(= 680-490 = 190\) |
\(\text{Outlier if \$970 is greater than:} \)
\(Q_3 + 1.5 x\times IQR = 680 + 1.5 \times 190 = \$965 \)
\(\therefore\ \text{The wage \$970 per week is an outlier.}\)
ii. \(\text{All values increase by \$20, but so too does the mean.} \)
\(\text{Therefore the spread about the new mean will not change} \)
\(\text{and therefore the standard deviation will remain the same.} \)
This set of data is arranged in order from smallest to largest.
`5, \ 6, \ 11, \ x, \ 13, \ 18, \ 25`
The range is six less than twice the value of `x`.
Which one of the following is true?
`D`
`5, 6, 11, x, 13, 18, 25`
| `text(Range)` | `= 2x – 6` |
| `25 – 5` | `= 2x – 6` |
| `2x` | `= 26` |
| `x` | `= 13` |
| `:.\ text(Median)` | `= 13` |
`Q_1 = 6\ \ \ \ \ Q_3 = 18`
`:.\ text(IQR) = 12`
`=> D`