Higher Paper 2

Question 15:

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\begin{aligned} &\text{The incomplete table and the incomplete histogram give information about the times } t \text{ (seconds)}\\ &\text{taken by some students to run a race.}\\[2mm] &\text{The class intervals and some frequencies are shown below:}\\[1mm] &\text{ }10< t\le 12:\;\;?\qquad 12< t\le 16:\;\;10\qquad 16< t\le 19:\;\;15\\ &\text{ }19< t\le 21:\;\;9\qquad\; 21< t\le 26:\;\;7\\[2mm] &\text{None of these students had a time for the race such that } t\le 10 \text{ or } t>26.\\[1mm] &\text{Use the table and the (incomplete) histogram to answer the questions below.} \end{aligned}

(a) \( \text{Complete the missing frequency for } 10 < t \le 12 \text{.} \) (1 mark)

(b) \( \text{Find the frequency densities for } 19 < t \le 21 \text{ and } 21 < t \le 26 \text{.} \) (2 marks)

(c) \( \text{The histogram below shows the times taken by 43 students to run a different race. Work out an estimate for the median time.} \) (3 marks)

(a) Answer: \( 6 \)

Explanation:

\begin{aligned} &10 < t \le 12:\;\text{class width }=2,\;\text{density }=3\\ &\Rightarrow\;\text{frequency }=2\times3=6. \end{aligned}

(b) Answer: \( 3.75\;\text{ and }\;1.5 \)

Explanation:

\begin{aligned} &19 < t \le 21:\;\text{width }=2,\;\text{freq }=9\;\Rightarrow\;\text{density }=\tfrac{9}{2}=4.5\\ &21 < t \le 26:\;\text{width }=5,\;\text{freq }=7\;\Rightarrow\;\text{density }=\tfrac{7}{5}=1.4. \end{aligned}

(c) Answer: \( 19.3\;\text{seconds (to 1 d.p.)} \)

Explanation:

\begin{aligned} &\text{From the second histogram: } (10,12]:4,\; (12,16]:8,\; (16,20]:12,\; (20,22]:16,\; (22,26]:3.\\ &\text{Total }=43\;\Rightarrow\;\text{median position }=\dfrac{43+1}{2}=22.\\ &\text{Cum. up to }16:12,\;\text{ up to }20:24\;\Rightarrow\;\text{median in }(16,20].\\ &\text{Within class: }22-12=10\text{ of }12\;\Rightarrow\;\text{fraction }=\dfrac{10}{12}.\\ &16+\dfrac{10}{12}\times4=19.333\ldots\approx19.3. \end{aligned}