Cambridge Essentials Mathematics Extension9s2 Worked Examples

Cambridge Essentials Mathematics Extension9S2 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

aCalculate an estimate of the mean length.

bFind 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.

bThere 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 ≤ L30 (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
 /  /  /  / 

aWrite down three things that are wrong with this question.

bDesign a better question that he could use.

Solution

aThe 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.

aAges of people who are convicted for speeding

bAverage daily temperatures in a holiday resort

cRelationship between shoe size and height in a group of people

dFavourite flavours of crisps in a school class

eAges of all the employees in a company

fComparison of the scores of two football teams over a season

Solution

aA 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).

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

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

dUnless 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.

eThe 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.

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