Brief Description

Pupils conduct a survey to investigate the ages of cars. They are encouraged to discover why the sample may be biased and are led to investigate official published figures to enable them to answer other questions about cars.

Design time: 4-5 hours

Aims and Objectives

On completion of this unit pupils should be able to conduct a simple counting survey and represent the results on a bar chart. They will have practised working with published data, reading tables, drawing histograms, calculating means and drawing simple inferences from tabulated data.

The pupils should be more aware of possible bias in sampling; how data are collected in different circumstances; some of the problems that arise in connection with data collection, sources of published data; and the difficulties that can arise in interpretation and comparison of statistics.

Prerequisites

Pupils need to be able to (i) convert fractions to decimals, (ii) plot points for a time series, (iii) use tally marks, (iv) draw a bar chart and (v) work with percentages and change fractions to decimals and percentages.

The optional section involves calculating a mean of a frequency distribution. It will help if they have met this technique before.

Equipment and Planning

Section A2 involves carrying out a survey of the last letters of car number plates, and this will need planning. Page R1 is used to record the results. The unit starts with a discussion of some of the questions that people may want to know about cars and uses the results of the survey to estimate some answers. Since the local survey is bound to be biased, as is shown by comparing the estimates from the local survey with the national figures available, time is taken to look at the effect of bias. Section C uses national figures to look at the number of new cars produced and to calculate the mean age of cars on the road and the scrapping rate. A final optional section estimates the age distribution of scrapped cars and calculates the mean life of a car.

Section B4 is optional and gives practice in estimation calculations first introduced in B1. C5 is a harder optional section for brighter pupils only. Individual harder questions are marked with an asterisk.

Calculators would be useful for Sections B and C. It is possible to interrupt the unit at the end of Section B and return later to Section C.

Detailed Notes

Section A

A1
The questions are designed to initiate a discussion. To find out how long cars last, the age at which the car is scrapped is required. The growth rate of the total number of cars can be determined from the total numbers licensed each year. A direct method is to use published data (in the Monthly Digest of Statistics). There is a time lag between the collection of data and publication (perhaps as much as two years). Records are now being kept centrally at the Driver and Vehicle Licensing Centre, Swansea. Answers can also be estimated from surveys which provide current data more quickly, but great care is needed, as shown in this unit.

A2
The system of indicating the year by the end registration letter should be discussed with the pupils. It was changed in 1967 after pressure from car manufacturers to boost mid-year sales. There should be fewer E registered than D registered cars, but this may not show up, because of small numbers. The letters I, and Q were omitted to avoid confusion with the numbers 1 and 0. Q is used for cars which have been registered but on which no tax has yet been paid.

To carry out the survey, each group of three pupils will need page R1 and an aid for counting up to 200 (eg. tallying or a list of numbers from 1 to 200). Decide in advance where to take the survey. There should be a reasonable traffic flow. It may be possible to carry out the survey for homework. If there is little traffic, results could be combined over several carefully chosen time-periods (to avoid double-counting). It is important that pupils carry out a survey to enable them to become aware of the ensuing problems. The pupils should be reminded about road safety prior to the survey. A car park could be used for further data.

Only private cars and vans are considered in the survey, because other vehicles, such as lorries or buses, have a different life-span. In 1975 the average annual distance travelled by cars was 8600 miles; buses and coaches travelled 26800 miles, whilst goods vehicles travelled 13700 miles (Transport Statistics 1965-75). Cars with foreign registration plates are not included because their age is not obvious. Cars with 'personalized' number plates are usually too few to have any significant effect, except perhaps at some rally: one would get new cars showing old registration plates.

A3
The result s given in Table 2 are to help pupils analyse their own results. The survey was carried out at 9 a.m. on one of the main roads leading into Rochdale. The distribution has few T registered cars because the survey was conducted at the end of August. The first class in the table 'F and earlier' may include some personalized plates. The first class in the pupils' table could be a later letter than F, depending on the current letter.

A4
Care is needed when combining separate samples. Results can only be combined if one can be fairly confident that a car will not have been included more than once. Perhaps only some results can be combined.

Section B

B1
The pupils calculate an estimate of the N registered cars in Britain. Part e will not apply if combined class results are used. Pupils should be made aware that all estimates are only approximate. These particular estimates will also be biased because of the method of sampling used.

Some pupils might find it easier to work with fractions (such as ¹⁷/₂₀₀) and cancel rather than with decimals.

B2
The results of the analysis clearly show the bias in the Rochdale sample. Further calculations for other letter registrations can be done.

B3
Pupils should write down their own answers to the questions, but a final class discussion would set these in perspective.

At 8 a.m. a higher proportion of newer company vehicles might be noticed. On early closing day the bias would be against the cars of local shoppers and/or the second car of the family (which may be older). These groups might feature more strongly in a sample on the town car park, which is less dependent on vehicle-traffic-hours. There would be a different distribution in a strategic highly priced park than a long-stay cheaper (or free) one, or the school car park. A sample outside a secondhand car centre would be biased towards older cars, while that in an area of expensive houses might contain a higher proportion of newer cars. On an industrial estate there are likely to be many business cars, which will be of recent registration. These would not feature to such an extent in a survey done on a Sunday. On motorways, representative of longer, faster journeys, newer cars would tend to predominate.

*B4
The survey was carried out on the Friday of a Bank Holiday weekend and provides the data for extra practice on previous work in the section. The number of touring caravans in Britain is an estimated figure, based on the numbers produced, their life span and use. The work assumes that touring caravans are not left on any one site or that those left on sites over the season will be towed by cars of the same age distribution. These factors can be discussed as sources of bias. Estimates can be obtained for other registration letters.

Section C

C1
Discussion on listing large numbers in terms of thousands in tables may be beneficial. A common error in reading such tables is to forget that the units are thousands. The early questions are simple, to help pupils famaliarize themselves with the table. They may require help in plotting the axis in e.

A more subtle point in the analysis of the time series is that the proportion of new cars can appear to fall even when the number of cars is still increasing, because it is expressed as a fraction of a growing fleet. Economic factors may also be brought into the discussion of trends. After a time there will probably be an upturn, because cars have a limited lifespan.

C2
Parts a and b may be omitted by the more able pupils. The column 'Average age' in Table 5 may need explaining. The rationale for the last part of the histogram may be expanded upon - it assumes an even distribution of cars over the 4-year period. The data are plotted (from 0 to 20 years) on the histogram in an opposite direction to that of the bar chart in A3 (from 'F and earlier' to current letter). If the sample were representative and the present distribution similar to that in 1977, one would get a mirror image.

If pupils have not used the mean of a distribution before, part f may be omitted (see Cutting it Fine or Seeing is Believing for work on the mean). The pupils may need help in calculating the mean.

The formula may be mentioned to brighter pupils who have already met the notation.

C3
The growth in the car population has been levelling off. This could be because of saturation or a poor economy, or both. Growth is calculated as a proportion of stock at the beginning of the year (and similarly in the next section for cars scrapped). Weaker pupils may prefer to omit part c.

C4
The scrapping rate has fluctuated over the years - it is also affected by economic upswing or depression. Weaker pupils may prefer to omit parts b and c.

*C5
This part may be found difficult by some pupils. All may require further explanation of Table 6. There are a number of hidden assumptions in the calculations, necessitated by lack of data. The year of the first registration is given only for two-year periods. The average ages taken in 1973 and 1975 are the mid-points of the class intervals. The results are based on a sample first taken in 1973, so it cannot include all the cars registered in that year; hence these have been omitted. The average age at scrapping is also the mid-point of an interval; a car registered in December 1972, but scrapped in 1973 could be '0' years old, while a car registered in January 1971, but scrapped in December 1974, would be 4 years old. 2 is taken as the mid-point.

The data refer only to private cars.

The histogram gives a visual impression of the lifetime of cars. The mean age is more difficult to ascertain but provides a definite figure.

Answers

A1		See detailed notes.
A2		See detailed notes.
A3		See detailed notes.
B1	f	See detailed notes.
B2	c	See detailed notes.
B3		See detailed notes.
B4	a	51200
	b	12800
	c	112000
C1	c	8247000, 11515000, 14047000
	d	1972,
	e	1964,
	f	1975
C2	a	2006 thousand; 7,
	b	1260 thousand, 11
	c	18 years; 137000
	e	See detailed notes.
	f	(2125), (8289), 14395, 14042, (16533), 13860, 8606, 3510, (4932). Total 86292 (thousand car-years) Mean age = 86292/14040 = 6.14 (6) years
C3	a	655000, 780000, 142000, 108000
	b	1975
	c	0.0543, 5.4%, 0.0613, 6.1%, 0.0105, 1.1%, 0.0079, 0.8%
C4	a	1008000, 865000, 1092000, 1059000
	b	0.0836, 8.4%, 0.068, 6.8%, 0.0809, 8.1%, 0.0776, 7.8%
	c	Smaller
C5	a	(11), 82, 172, (442), 651, 364, (512)
	b	Mean age = 23988/2234 = 10.7 years

References

Monthly Digest of Statistics (Central Statistical Office)
Transport Statistics in Great Britain (Dept. of Transport)
Vital Statistics about the Caravan Industry (National Caravan Council)

Page R1

Registration letter	Tally marks	Total
No letter
A
B
C
D
E
F
G
H
J
K
L
M
N
P
R
S
T
V
W
X
Y
Z
Total

Date:
Time:
Place:

Test Questions

1. Give one reason for conducting a vehicle survey.
2. Table T1 shows the results of a car survey. It was taken on a Friday morning, 10.30 a.m. to 11.30 a.m., on a city ring road.
   

   End registration letter	None	A	B	C	D	E	F	G	H	J	K	L	M	N	P	R	S	Total
   Number of cars	4	1	0	2	1	2	2	3	6	9	15	24	26	29	30	32	14	200

   Table T1 - Car survey results, 20 January, 1978

   1. How many cars had no end letter?
   2. Which was the first letter to have more registration than this?
   3. Copy and complete:
      Half the cars were registered in the year of the letter _____ or later.
3.  
   1. What is the advantage of adding results from surveys done in different places?
   2. One group of pupils did a car survey at one end of a road; another group did a survey at the other end of the road. They combined their results. Why might this be misleading?
4.  
   1. What is the fraction of H-registered cars in Table T1?
   2. There were approximately 15 million cars licensed in January 1978. Use your answer to 4a to estimate the number of these that were H-registered.
5. Notice when and where the survey in question 2 was carried out.
   1. Which of the following would you expect to feature most in the survey?
      Shopping traffic
      Business cars
      Holiday traffic
      New deliveries to garages

   How would you expect the results to differ:

   2. On a free town centre car park, again on Friday morning?
   3. On a coast road during a Sunday afternoon in summer?
6. Table T2 gives the number of new buses compared to all buses licensed that year.
   

   Year	Licences current (thousands)	First registered
   		Number	Percentage
   1965	81.7	5474	6.7
   1966	78.5	5399	6.9
   1967	78.8	5007	6.4
   1968	79.7	5135	6.4
   1969	79.1	5134	6.5
   1970	77.8	5018	6.4
   1971	18.1	6213	8.0
   1972	76.7	6440	8.4
   1973	78.8	7177	9.2
   1974	78.6	5220	6.6
   1975	79.6	5481	6.9

   Table T2 - Buses licensed and first registered 1965-75
   (Source: Transport Statistics, Great Britain, 1965-75)

   1. In which years were new buses more than 7% of the total licensed that year?
   2. In which year were new buses less than 6'5% of the total licensed that year?
   3. How many buses were there in 1966; in 1967?
   4. How many more buses were there in 1967 than in 1966?
   5. Write the increase in buses in 1967 (from 1966) as a fraction of the number of buses in 1966 (use your answers to c and d)
   6. How many more buses were there in 1975 than in 1974?
   7. How many new buses were there in 1975?
   8. How many buses were scrapped from 1974 to 1975?
   9. What fraction of the 1974 buses were scrapped between 1974 and 1975?
7. A survey of shoppers is made between 10.30 and 11.30 on a Tuesday morning. The sample contains 85% women. What difference would you expect to find in a survey taken during the same hours on a Saturday?

Answers

1		To determine the ages of vehicles and related ideas, e.g. scrapping rate, growth rate, in the area where the survey is carried out
2	a	4
	b	H
	c	N
3	a	To increase the sample size
	b	Most of the cars would have been included twice; the surveys are ngt independent.
4	a	3/100 (6/200)
	b	450000
5	a	Business cars
	b	More shoppers' cars, which would tend to be older vehicles
	c	Fewer business vehicles; relatively older cars would be on the road
6	a	1971, 1972, 1973
	b	1967, 1968, 1970
	c	78500, 78800
	d	300
	e	3/785
	f	1000
	g	5481
	h	4481
	i	4481/78600
7		It would contain a smaller percentage of women (and a greater percentage of men).

Connections with Other Published Units from the Project

Other Units at the Same Level (Level 3)

Net Catch
Phoney Figures
Cutting it Fine
Pupil Poll
Multiplying People

Units at Other Levels In the Same or Allied Areas of the Currlculum

Level 1

If at first...
Tidy Tables
Leisure for Pleasure

Level 2

Opinion Matters

Level 4

Figuring the Future
Retail Price Index
Sampling the Census
Equal Pay

This unit is particularly relevant to: Social Sciences, Mathematics.

Interconnections between Concepts and Techniques Used In these Units

These are detailed in the following table. The code number in the left-hand column refers to the items spelled out in more detail in Chapter 5 of Teaching Statistics 11-16.

An item mentioned under Statistical Prerequisites needs to be covered before this unit is taught. Units which introduce this idea or technique are listed alongside.

An item mentioned under Idea or Technique Used is not specifically introduced or necessarily pointed out as such in the unit. There may be one or more specific examples of a more general concept. No previous experience is necessary with these items before teaching the unit, but more practice can be obtained before or afterwards by using the other units listed in the two columns alongside.

An item mentioned under Idea or Technique Introduced occurs specifically in the unit and, if a technique, there will be specific detailed instruction for carrying it out. Further practice and reinforcement can be carried out by using the other units listed alongside.

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