Cambridge Essentials Mathematics Extension 9 S2 Worked Examples

Cambridge Essentials Mathematics Extension 9 S2 Worked examples

/ S2 Worked examples

Question 1

The grouped frequency table shows the length in centimetres of 40 straws in a box.

Length, L (cm) / Frequency
20 ≤ L < 25 / 8
25 ≤ L < 30 / 3
30 ≤ L < 35 / 7
35 ≤ L < 40 / 7
40 ≤ L < 45 / 15

a Calculate an estimate of the mean length.

b Find the class interval in which the median lies.

Solution

a / Length, L (cm) / Frequency / Midpoint / Frequency ´
midpoint
20 ≤ L < 25 / 8 / 22.5 / 180
25 ≤ L < 30 / 3 / 27.5 / 82.5
30 ≤ L < 35 / 7 / 32.5 / 227.5
35 ≤ L < 40 / 7 / 37.5 / 262.5
40 ≤ L < 45 / 15 / 42.5 / 637.5
Total / 40 / 1390

The estimated mean is

The estimated mean of the straw lengths is 34.75 cm.

b There are 40 straws, so the median length is halfway
between the 20th and 21st lengths.
The first 8 lengths are in the interval 20 ≤ L < 25,
the next 3 are in the interval 25 ≤ L < 30 (11 so far),
the next 7 are in the interval 30 ≤ L < 35 (18 so far).
The 20th and 21st straws are in the interval 35 ≤ L < 40.
So the median length lies within the interval 35 ≤ length (cm) < 40.

Question 2

Vincent is investigating how much television people watch.

He used this question to collect data.

How much television do you watch?
1–2
hours / 2–4
hours / 4–6
hours / 6–8
hours / 8–10
hours
£ / £ / £ / £ / £

a Write down three things that are wrong with this question.

b Design a better question that he could use.

Solution

a The question assumes that everyone watches television; some may not.
The question is too vague: it is not clear what time period it covers.
The response boxes overlap, and do not cover all possible answers.

b / How many hours in the last week did you watch television?
Less than 1 hour / 1–9
hours / 10–19 hours / 20–29 hours / 30–39 hours / More than 40 hours
£ / £ / £ / £ / £ / £

Question 3

Look at each of these surveys and suggest the best way to display the information that might be collected.

a Ages of people who are convicted for speeding

b Average daily temperatures in a holiday resort

c Relationship between shoe size and height in a group of people

d Favourite flavours of crisps in a school class

e Ages of all the employees in a company

f Comparison of the scores of two football teams over a season

Solution

a A bar chart would be appropriate, to show the distribution by age and the different numbers involved. A double or split bar chart could be used, to separate the results by gender (male or female).

b A line graph would be appropriate, to show the trend or how the temperature varies according to season.

c This would best be represented in a scatter graph, as it is seeking a relationship.

d Unless this was to be used as a buying list for a school disco or similar event, these results would be shown in a pie chart, which would demonstrate proportions without necessarily giving numbers.

e The ages of a large number of people can be represented in a stem-and-leaf diagram, since they will fall easily into groups differentiated by multiples of ten.

f A box plot is ideal for comparing distributions as it quickly shows the spread of the data, the median and the extreme or end values.