(a)

\begin{align*} \boxed{2 \phantom{0} 4 \phantom{0} 5} \phantom{.} & 6 \phantom{.} \boxed{7 \phantom{0} 8 \phantom{0} 10} \\ \\ \text{Range} & = \text{Maximum} - \text{Minimum} \\ & = 10 - 2 \\ & = 8 \\ \\ \text{Lower quartile} & = 4 \\ \\ \text{Median} & = 6 \\ \\ \text{Upper quartile} & = 8 \\ \\ \text{Interquartile range} & = 8 - 4 \\ & = 4 \end{align*}

(b)

\begin{align*} \boxed{51 \phantom{0} 54 \phantom{0} 63 \phantom{0} 64} & \boxed{66 \phantom{0} 70 \phantom{0} 72 \phantom{0} 80} \\ \\ \text{Range} & = 80 - 51 \\ & = 29 \\ \\ \text{Lower quartile} & = {54 + 63 \over 2} \\ & = 58.5 \\ \\ \text{Median} & = {64 + 66 \over 2} \\ & = 65 \\ \\ \text{Upper quartile} & = {70 + 72 \over 2} \\ & = 71 \\ \\ \text{Interquartile range} & = 71 - 58.5 \\ & = 12.5 \end{align*}

(c)

\begin{align*} \boxed{ 9 \phantom{0} 10 \phantom{0} 14 \phantom{0} 16 } \phantom{.} & 18 \phantom{.} \boxed{22 \phantom{0} 27 \phantom{0} 32 \phantom{0} 40} \\ \\ \text{Range} & = 40 - 9 \\ & = 31 \\ \\ \text{Lower quartile} & = {10 + 14 \over 2} \\ & = 12 \\ \\ \text{Median} & = 18 \\ \\ \text{Upper quartile} & = {27 + 32 \over 2} \\ & = 29.5 \\ \\ \text{Interquartile range} & = 29.5 - 12 \\ & = 17.5 \end{align*}

(d)

\begin{align*} \boxed{86 \phantom{0} 98 \phantom{0} 102 \phantom{0} 138 \phantom{0} 164 } & \boxed{168 \phantom{0} 184 \phantom{0} 207 \phantom{0} 244 \phantom{0} 250 } \\ \\ \text{Range} & = 250 - 86 \\ & = 164 \\ \\ \text{Lower quartile} & = 102 \\ \\ \text{Median} & = {164 + 168 \over 2} \\ & = 166 \\ \\ \text{Upper quartile} & = 207 \\ \\ \text{Interquartile range} & = 207 - 102 \\ & = 105 \end{align*}

(e)

\begin{align*} \boxed{2.7 \phantom{0} 4.9 \phantom{0} 6.7 \phantom{0} 8.5} \phantom{.} & 10.4 \phantom{.} \boxed{11.8 \phantom{0} 13.1 \phantom{0} 15.1 \phantom{0} 22.4} \\ \\ \text{Range} & = 22.4 - 2.7 \\ & = 19.7 \\ \\ \text{Lower quartile} & = {4.9 + 6.7 \over 2} \\ & = 5.8 \\ \\ \text{Median} & = 10.4 \\ \\ \text{Upper quartile} & = {13.1 + 15.1 \over 2} \\ & = 14.1 \\ \\ \text{Interquartile range} & = 14.1 - 5.8 \\ & = 8.3 \end{align*}

(i)

\begin{align*} \boxed{0 \phantom{0} 0 \phantom{0} 1 \phantom{0} 6 \phantom{0} 6} & \boxed{ 9 \phantom{0} 9 \phantom{0} 24 \phantom{0} 27 \phantom{0} 29} \\ \\ \text{Median} & = {6 + 9 \over 2} \\ & = 7.5 \\ \\ \text{Lower quartile} & = 1 \\ \\ \text{Upper quartile} & = 24 \end{align*}

(ii)

\begin{align*} \text{Range} & = \text{Maximum} - \text{Minimum} \\ & = 29 - 0 \\ & = 29 \\ \\ \text{Interquartile range} & = \text{Upper quartile} - \text{Lower quartile} \\ & = 24 - 1 \\ & = 23 \end{align*}

(i)

\begin{align} \text{Median position} & = {20 + 1 \over 2} \\ & = 10.5 \\ \\ \text{Median} & = {42 + 48 \over 2} \\ & = 45 \end{align}

(ii)

\begin{align*} \text{Range} & = \text{Maximum mark} - \text{Minimum mark} \\ & = 95 - 9 \\ & = 86 \end{align*}

(iii) The lower quartile is the median of the first 10 students while the upper quartile is the median of the next 10 students.

\begin{align*} \text{Interquartile range} & = \text{Upper quartile} - \text{Lower quartile} \\ & = {73 + 79 \over 2} - {28 + 30 \over 2} \\ & = 76 - 29 \\ & = 47 \end{align*}

(a)(i)

\begin{align*} \text{Median} & = \$ 97 \phantom{00000} [\text{At 150 mark}] \\ \\ \text{Lower quartile} & = \$ 88 \phantom{00000} [\text{At 75 mark}] \\ \\ \text{Upper quartile} & = \$ 105 \phantom{00000} [\text{At 225 mark}] \end{align*}

(a)(ii)

\begin{align*} \text{Interquartile range} & = \text{Upper quartile} - \text{Lower quartile} \\ & = 105 - 88 \\ & = 17 \end{align*}

(b)(i)

\begin{align*} 300 \times 20\% & = 60 \\ \\ \text{20th percentile of daily earnings} & = \$ 85 \end{align*}

(b)(ii)

\begin{align*} 300 \times 90\% & = 270 \\ \\ \text{90th percentile of daily earnings} & = \$ 110 \end{align*}

(i)

\begin{align*} \text{Median height} & = 50 \text{ cm} \phantom{00000} [\text{At 28 mark}] \end{align*}

(ii)

\begin{align*} \text{Upper quartile} & = 57 \text{ cm} \phantom{00000} [\text{At 42 mark}] \end{align*}

(iii)

\begin{align*} \text{Lower quartile} & = 39 \text{ cm} \phantom{00000} [\text{At 14 mark}] \end{align*}

(iv)

\begin{align*} \text{No. of plants having heights less than or equals to 57 cm} & = 42 \\ \\ \text{No. of plants having heights more than 57 cm} & = 56 - 42 \\ & = 14 \end{align*}

(a)(i)

\begin{align*} \text{Median length} & = 35.5 \text{ mm} \end{align*}

(a)(ii)

\begin{align*} \text{Upper quartile} & = 38.5 \text{ mm} \\ \\ \text{Lower quartile} & = 32.5 \text{ mm} \\ \\ \text{Interquartile range} & = 38.5 - 32.5 \\ & = 6 \text{ mm} \end{align*}

(b)

\begin{align*} \text{No. of healthy leaves} & = 600 \times 65\% \\ & = 390 \\ \\ \text{No. of leaves with length shorter than or equals to } h \text{ mm} & = 600 - 390 \\ & = 210 \\ \\ \text{From the graph, } h & = 34 \end{align*}

(a)(i)

\begin{align*} \text{Median mark} & = 23.5 \end{align*}

(a)(ii)

\begin{align*} \text{Upper quartile} & = 26.5 \end{align*}

(a)(iii)

\begin{align*} \text{Lower quartile} & = 20 \\ \\ \text{Interquartile range} & = 26.5 - 20 \\ & = 6.5 \end{align*}

(a)(iv)

\begin{align*} \text{No. of participants who scored less than 26 marks} & = 56 \\ \\ \text{No. of participants who scored less than 30 marks} & = 72 \\ \\ \text{No. of participants who scored more than or equal to 26 marks but less than 30 marks} & = 72 - 56 \\ & = 16 \end{align*}

(b)

\begin{align*} \text{No. of students who passed} & = 80 \times 37.5 \% \\ & = 30 \\ \\ \text{No. of students who failed} & = 80 - 30 \\ & = 50 \\ \\ \text{From the graph, passing mark} & = 25 \end{align*}

(a)(i)

\begin{align*} \text{Median travelling expenses (School } A) & = 42 \text{ cents} \end{align*}

(a)(ii)

\begin{align*} \text{Median travelling expenses (School } B) & = 58 \text{ cents} \end{align*}

(b)(i)

\begin{align*} \text{Upper quartile} & = 56 \text{ cents} \\ \\ \text{Lower quartile} & = 30 \text{ cents} \\ \\ \text{Interquartile range} & = 56 - 30 \\ & = 26 \text{ cents} \end{align*}

(b)(ii)

\begin{align*} \text{Upper quartile} & = 72 \text{ cents} \\ \\ \text{Lower quartile} & = 48 \text{ cents} \\ \\ \text{Interquartile range} & = 72 - 48 \\ & = 24 \text{ cents} \end{align*}

(c)

\begin{align*} \text{80% of students} & = 800 \times 80\% \\ & = 640 \\ \\ \text{80th percentile of travelling expenses (School } B) & = 76 \text{ cents} \end{align*}

(d)

\begin{align} & \text{Students from School B spend more on daily traveling expenses} \\ & \text{as the median traveling expense is higher (58 cents)} \end{align}

(i)

\begin{align*} \text{Lower quartile} & = 21 \\ \\ \text{Median} & = 28 \\ \\ \text{Upper quartile} & = 34 \end{align*}

(ii)

$$ 38 \text{ students} $$

(iii)

\begin{align*} \text{Lower quartile} & = 20 \\ \\ \text{Upper quartile} & = 28.5 \\ \\ \text{Interquartile range} & = 28.5 - 20 \\ & = 8.5 \end{align*}

(iv)

\begin{align*} \text{No. of students who scored less than or equals to 38} & = 37 \\ \\ \text{No. of students who received a gold award} & = 38 - 37 \\ & = 1 \\ \\ \text{% of students who received a gold award} & = {1 \over 38} \times 100 \\ & = 2.63157 \\ & \approx 2.63 \% \end{align*}

(v)

\begin{align} & \text{Class A performed better as the median score is higher.} \\ & \text{However, the results of Class A are less consistent as the interquartile range is larger.} \end{align}

(a)(i)

\begin{align*} \text{Median mark} & = 50 \end{align*}

(a)(ii)

\begin{align*} \text{70% of students} & = 500 \times 70\% \\ & = 350 \\ \\ \text{70th percentile} & = 60 \text{ marks} \end{align*}

(a)(iii)

\begin{align*} \text{Upper quartile} & = 62 \\ \\ \text{Lower quartile} & = 33 \\ \\ \text{Interquartile range} & = 62 - 33 \\ & = 29 \end{align*}

(a)(iv)

\begin{align*} \text{No. of cadets who scored less than 43 marks} & = 195 \end{align*}

(a)(v)

\begin{align*} \text{No. of cadets who passed} & = 500 \times 60\% \\ & = 300 \\ \\ \text{No. of cadets who failed} & = 500 - 300 \\ & = 200 \\ \\ \text{From the graph, passing mark} & = 44 \end{align*}

(b)

\begin{align*} \text{No. of cadets who scored distinctions in School } A & = 500 - 430 \\ & = 70 \\ \\ \text{Percentage of cadets who scored distinctions in School } A & = {70 \over 500} \times 100 \\ & = 14 \% \\ \\ \\ \text{No. of cadets who scored distinctions in School } B & = 500 - 355 \\ & = 145 \\ \\ \text{Percentage of cadets who scored distinctions in School } B & = {145 \over 500} \times 100 \\ & = 29 \% \end{align*}

(c)

\begin{align} & \text{I agree. Cadets from School B performed better in general as the median mark is higher.} \end{align}

(a)(i)

\begin{align*} \boxed{ 16 \phantom{0} 21 \phantom{0} 23 \phantom{0} 37 \phantom{0} 50 } & \boxed{53 \phantom{0} 65 \phantom{0} 80 \phantom{0} 81 \phantom{0} 100} \\ \\ \text{Range} & = \text{Maximum} - \text{Minimum} \\ & = 100 - 16 \\ & = 84 \end{align*}

(a)(ii)

\begin{align*} \text{Median} & = {50 + 53 \over 2} \\ & = 51.5 \end{align*}

(a)(iii)

\begin{align*} \text{Interquartile range} & = 80 - 23 \\ & = 57 \end{align*}

(b)(i)

\begin{align*} \boxed{79 \phantom{0} 99 \phantom{0} 103 \phantom{0} 114 \phantom{0} 121} & \boxed{171 \phantom{0} 198 \phantom{0} 200 \phantom{0} 235 \phantom{0} 308 } \\ \\ \text{Range} & = \text{Maximum} - \text{Minimum} \\ & = 308 - 79 \\ & = 229 \end{align*}

(b)(ii)

\begin{align*} \text{Median} & = {121 + 171 \over 2} \\ & = 146 \end{align*}

(b)(iii)

\begin{align*} \text{Interquartile range} & = 200 - 103 \\ & = 97 \end{align*}

(c)

\begin{align} & \text{City Y’s data show a greater spread as the interquartile range is larger.} \end{align}

(d)

\begin{align} & \text{In general, the air quality in City X is better.} \\ \\ & \text{The median PSI of 51.5 is lower than of City Y.} \\ & \text{City X’s PSI is less spread out since the interquartile range of 97 is lower,} \\ & \text{which suggests City X’s PSI readings are more consistent and lower.} \end{align}

(a)(i)

\begin{align*} \text{Lower quartile} & = 10 \text{ mins} \\ \\ \text{Median} & = 13 \text{ mins} \\ \\ \text{Upper quartile} & = 15.25 \text{ mins} \end{align*}

(a)(ii)

\begin{align*} \text{Interquartile range} & = 15.25 - 10 \\ & = 5.25 \text{ mins} \end{align*}

(b)

\begin{align*} \text{No. of clients who waited not more than 15 mins} & = 60 - 16 \\ & = 44 \\ \\ \text{% of clients who waited not more than 15 mins} & = {44 \over 60} \times 100\% \\ & = 73{1 \over 3} \% \end{align*}

(c)

\begin{align} & \text{The intersection represent the median waiting time (13 mins).} \\ \\ & \text{It is the point where the number of clients with waiting times more than t minutes (first curve)} \\ & \text{is equal to the number of clients with waiting times less than or equal to t minutes (second curve).} \end{align}