The time taken, $t$ minutes, to complete a puzzle was recorded for each of 150 students. These times are summarised in the table. Time taken ($t$ minutes) $t \leqslant 25$ $t \leqslant 50$ $t \leqslant 75$ $t \leqslant 100$ $t \leqslant 150$ $t \leqslant 200$ --- --- --- --- --- --- --- Cumulative frequency 16 44 86 104 132 150 (a) Draw a cumulative frequency graph to illustrate the data. (b) Use your graph to estimate the 20th percentile of the data.

Mathematics 9709/53 — May/June 2022

Cambridge AS Level · Probability & Statistics 1 · worked solutions for every part, with the mark scheme

questions

marks

minutes

Topics Representation of Data · Discrete Random Variables · Probability · The Normal Distribution · Permutations and Combinations

Q1Representation of DataFree sample

The time taken, $t$ minutes, to complete a puzzle was recorded for each of 150 students. These times are summarised in the table.

Time taken ( $t$ minutes)	$t \leqslant 25$	$t \leqslant 50$	$t \leqslant 75$	$t \leqslant 100$	$t \leqslant 150$	$t \leqslant 200$
Cumulative frequency	16	44	86	104	132	150

(a)

Draw a cumulative frequency graph to illustrate the data.

Fig. 1

DifficultyMedium-Easy

Worked solution

Approach

To draw a cumulative frequency graph, plot the cumulative frequency values against the corresponding upper class boundaries. The curve must start at the origin $(0,0)$ and be drawn as a smooth S-shaped curve.

Working

The data points to plot are:

$(0, 0)$
$(25, 16)$
$(50, 44)$
$(75, 86)$
$(100, 104)$
$(150, 132)$
$(200, 150)$

Set up the axes with time $t$ (in minutes) on the horizontal axis from 0 to 200, and cumulative frequency on the vertical axis from 0 to 150 (or 160 for clarity). Plot the points and join them with a smooth curve.

Answer

Cumulative frequency graph plotted with points $(0,0)$ , $(25,16)$ , $(50,44)$ , $(75,86)$ , $(100,104)$ , $(150,132)$ , $(200,150)$ joined by a smooth curve.

Final answer

Cumulative frequency graph plotted with points (0,0), (25,16), (50,44), (75,86), (100,104), (150,132), (200,150) joined by a smooth curve.

Detailed explanation

Walkthrough

To construct a cumulative frequency graph, we first identify the upper class boundaries and their corresponding cumulative frequencies. The data given provides cumulative frequencies for $t \leqslant 25, 50, 75, 100, 150, 200$ . We also include the implicit starting point $(0, 0)$ since no students finished in less than 0 minutes. We plot these seven points on a graph with time on the horizontal axis and cumulative frequency on the vertical axis. Finally, we join the points with a smooth, increasing S-shaped curve (ogive).

Key Takeaways

A cumulative frequency graph (ogive) is plotted using upper class boundaries on the horizontal axis and cumulative frequencies on the vertical axis.
The curve always starts at the origin $(0,0)$ and ends at the total frequency $(\text{max time}, N)$ .
The curve must be smooth and monotonically increasing.

Common Mistakes

Forgetting to include the origin $(0,0)$ as the starting point.
Joining the points with straight line segments instead of a smooth curve.
Using incorrect axes labels or scales.

Things to Be Careful About

Ensure the vertical axis scale accommodates the total frequency (150), so going up to 160 is a good practice.
The horizontal axis must cover the maximum time value (200).
The curve must be smooth; jagged lines will not be awarded marks.

Techniques used

plot cumulative frequency pointsdraw smooth cumulative frequency curve

(b)

Use your graph to estimate the 20th percentile of the data.

DifficultyMedium-Easy

Worked solution

Approach

The 20th percentile corresponds to the value of $t$ where the cumulative frequency is $\frac{20}{100}$ of the total frequency. Calculate this cumulative frequency value, then read the corresponding $t$ from the graph.

Working

Total frequency $N = 150$ .
Cumulative frequency for the 20th percentile:

\frac{20}{100} \times 150 = 30

Draw a horizontal line from cumulative frequency $= 30$ to the curve, then a vertical line down to the horizontal axis to read the value of $t$ .

From the graph, the value of $t$ is approximately 40 minutes (acceptable range: 37.5 to 42 minutes).

Answer

Final answer

Detailed explanation

Walkthrough

To find the 20th percentile, we first determine what cumulative frequency corresponds to 20% of the data. Since there are 150 students in total, we calculate $\frac{20}{100} \times 150 = 30$ . This means the 20th percentile is the time taken by the 30th student when ordered from fastest to slowest. We then locate 30 on the vertical cumulative frequency axis, draw a horizontal line to intersect the curve, and drop a vertical line down to the horizontal time axis to read the estimated time $t$ . Reading from the graph, this value falls between 37.5 and 42 minutes.

Key Takeaways

The $p$ -th percentile of $N$ data points is found at cumulative frequency $\frac{p}{100} \times N$ .
Percentiles are read from a cumulative frequency graph by moving horizontally from the frequency axis to the curve, then vertically to the variable axis.

Common Mistakes

Using the wrong formula for the percentile position (e.g., using $\frac{p}{100} \times (N+1)$ which is for discrete data, not cumulative frequency graphs).
Reading the graph incorrectly, such as reading the cumulative frequency instead of the time $t$ .
Forgetting to draw the construction lines from the axes to the curve.

Things to Be Careful About

The mark scheme allows a range of acceptable answers (37.5 to 42) due to the nature of reading from a hand-drawn graph.
Ensure the graph used is increasing; if the curve is drawn incorrectly in part (a), the answer in (b) will be marked wrong (B1 FT).

Techniques used

calculate percentile valueread value from cumulative frequency graph

The rest of this paper

6 more questions

Q2Representation of Data3M
Q3Discrete Random Variables6M
Q4Discrete Random Variables · Probability7M
Q5The Normal Distribution10M
Q6Probability10M
Q7Permutations and Combinations11M

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