Graph paper and rulers (preferably transparent) are required. Calculators and tracing paper may be useful.

This sets the scene for the unit and can be used for an opening discussion. The idea is to study changes over a period of years and to use the figures to make predictions about future events.

You could discuss with pupils which figures are most appropriate to build up a statistical picture of their lifetimes and how the relevant data might be obtained. Much of the data will be found in the Annual Abstract of Statistics or Social Trends. Other interesting topics not mentioned in the unit include sales of LP and singles records, and the number of motorcycle registrations.

The figures discussed here concern television, which was a luxury 30 years ago. Now over 95% of households have a set, and colour television is becoming increasingly popular.

B1

A simple method of prediction, using the mean annual increase, is given.

d Using the mean annual increase gives an estimated figure for 1975 of:
173 + (171 — 136)/8  million = 17.8 million

e No. The licence fee is paid per household.

Estimates for Great Britain, in 1971, gave 18 million households with an average household size of 3.

Notice that the licences refer to the UK and that the figures show neither the number of television owners who have no licence nor the number of households with more than one television set.

B2

a Pupils may need help in choosing scales for the graph. Comparison of lines used for prediction in f is easier if all pupils use the same scale. Lines can then be traced and overlaid on a neighbour’s axes.
c The points lie approximately in a straight line, and the line can be extended to allow prediction.
e Pupils use interpolation to check if the line they have drawn gives a reasonable answer.
f Estimates obtained from the linear regression line give:
          1975-18.0 million;     1977-18.9 million;     1985 - 22.4 million

Clearly pupils’ results may vary considerably from these estimates, and the variations between pupils’ estimates can make useful discussion points on ways of improving the line. Lines can be compared by using tracing paper.

B3

Pupils should be made aware that the farther they are away from known data when they predict, the greater is the chance of larger error.

c It is likely that a ‘ceiling’ will be reached, unless the number of households continues to increase, since the number of licences is limited to the number of households.

*B4

This optional section investigates upper and lower limits on prediction, using the greatest and smallest known annual increases.

a  The largest annual increase is 0.8 million (1967-1968), so the highest number likely in 1975 is:
                17.3 + 0.8 million =18.1 million

b Using the same annual increases, and using 1975 limits to predict 1976, and 1976 to predict 1977 gives:

1976: lowest limit l7.4 + 0.1 million = 17.5 million
highest limit 18.1 + 0.8 million =18.9 million

1977: lowest limit 17.5 + 0.1 million = 17.6 million
highest limit 18.9 + 0.8 million = 19.7 million

Notice that the range between the limits in 1975 was 0.77 million, in 1976 was 1.4 million and in 1977 was 2.1 million. Clearly, the longer the period over which prediction is made, the greater is the range and the less reliable the prediction.

This introduces the need to update predictions every time further data are available.

An actual value could lie outside the predicted range. This happened in
1976. Using the true 1975 figure, the estimated range for 1976 would be:

17.7 + 0.1 million to 17.7 + 0.8 million
= 17.8 million to 18.5 million

In fact the actual value for 1976 was 177 million

*B5

This optional section is for brighter pupils or can be done by faster working pupils in a mixed ability situation, while others catch up. It can be omitted without destroying the thread of the unit.

Not all data approximate successfully to a straight line, and this section gives pupils a chance to fit smooth curves ‘by eye’ and to use the curves to make predictions.

Later years give (in millions):

This indicates that colour licences exceeded black and white in the period 1976-1977.

Pupils are now expected to improve their attempts to draw lines of best fit by ensuring that they pass through the bivariate mean in addition to passing as close to the other points as possible.

Cl
This explains how to calculate and plot the bivariate mean, which occurs at (1970, 204.3). Pupils should be encouraged to use the same scales to allow comparisons in the optional question h. Suggested scales are 1 cm for 1 year on the horizontal axis and 2 cm for 50 million on the vertical axis.

e It is hoped that pupils will feel intuitively that the line should pass through the central point, since it is a point which takes every point into account.

g Estimates from the least squares regression line give:
1976 — 81.5 million
1977 — 61.0 million

Predictions are unlikely to be as accurate as these, of course.

*h This is for faster pupils, while the slower ones catch up. It can be omitted without destroying the thread of the unit.

C2
Price increases are often percentage increases, rather than absolute increases.

It is not, therefore, expected that a straight line increase should fit these data well. Pupils may comment on this fact when they try to draw the line. This is followed up in the optional C3.

b The bivariate mean is (1970, 33.8).

Estimates from the least squares regression line give:
d l976—57.lp
1977 — 61.0p

Answers within a reasonable range of these would be acceptable.

e The attendance line has a negative slope; the admission charges line a positive slope. An opportunity exists here to discuss the relation between costs and revenue and break-even points.

Possible outcomes are lower attendances -increased charges.

Lower prices would lead to higher attendances, but these are unlikely to be sufficient to offset the effects of inflation on staff wages, etc.

f The additional data show the line to be even less suitable as a predictor.

Despite the steep rise in prices, the attendances have not dropped dramatically over the same period. Mention could be made of the rise in average wage rates over the same period.

*C3

This is optional for those pupils who did B5. and follows on from C2.

The figures approximate quite well to an exponential curve. It is expected that the estimates from the curve will match more closely to the true values than those obtained from the line.

*C4

This gives optional reinforcement on calculating bivariate means, and adjusting trend lines to pass through this central point. Useful comparisons can then be made of original estimates with estimates obtained from the new trend line.

It is possible that their original estimates may be nearer the real figures than their new ones. This can be discussed. It should not be allowed to discredit values obtained from a line through the bivariate mean, but does illustrate the inapplicability of fitting straight lines to data which have a limiting ceiling.

*Section D

This is optional throughout. Page R1 will be required if pupils are not expected to make copies of Tables 4 and 5.

Dl
Deviations given in Table 4 are those obtained from the least squares regression line drawn in Figure 2. Pupils should not be expected to understand how to draw this line.

e Adding all the deviations should produce zero. Rounding errors may affect the total, however. Other suggestions include looking for the line giving the smallest maximum absolute deviation, or the smallest sum of the absolute deviations.

D2
No justification is given for this method here. It is meant as a little light relief to see who can produce the smallest sum of squares. The least squares method is widely used, since if the underlying distribution is normal, the least squares method gives the maximum likelihood estimates for the parameters of the line. The total 1328 is from the least squares regression line, with deviations rounded to integers; it is unlikely to be bettered

This is mainly for reinforcement, but does include the idea of plotting two sets of data on the same axes, using different vertical axes, and optional updating of lines in the light of further information.

El
a Telephones are cheaper and quicker to use, and get an instant reply.

b 1 Telegrams produce a written record of the message.

2 Telegrams can be read when a person arrives at the office; telephone calls require both people to be available simultaneously.

c The GPO need information for updating equipment, training, etc.

E2
Telephone calls are recorded in billions. In government publications a billion is a thousand million (10⁹).
b The centre of the time period is midway between 1969 and 1970.

E3
Estimates from the least squares regression line give:

c The data for telegrams fit a straight line far less well than those for telephone calls, which may affect the accuracy of predictions.

*g New means, including 1975, 1976 and 1977 figures are:
Telegrams — 7.3 million; telephone calls — 11.3 billion

The centre of the time period is 1971.

Estimates for 1985 from the new regression line give:
Telegrams — 2.5 million, telephone calls — 20.9 billion.

j Sudden changes cannot be foreseen, yet may drastically affect future events, so up-to-date data must be used wherever possible.

E4
a, b   Encourage pupils to think of possible ideas for the changing fortunes of telegrams and telephones.
c Telex systems have the advantage of using the speed of telephone lines, while giving a permanent record.

B3 c see detailed notes

*B4 a 0.8 million (1967-8); 18.1 million b see detailed notes

C1 2247/11 = 204.3

C2 c 372/11 = 33.8 or 34p
Dl e 0 f See detailed notes. g See detailed notes.
E2 b Telegrams: 8l.6 million/10 = 8.2 million
Telephone calls: 97.7 billion/10 = 9.8 billion

1 Give two possible meanings for 18 passenger-kilometres.

2 a By which method was travel decreasing during the 20 years?
b By which methods was travel increasing during the 20 years?

3 On graph paper prepare a vertical scale from 0 to 100 billion passenger-km, and a horizontal scale from 1955 to 1995.
a Use + signs to plot the rail figures from Table Tl.
b Draw a straight line close to the + points, showing the trend in rail travel.
c Use o signs to plot the bus figures from Table Tl.
d Calculate the arithmetic mean of the five figures of bus travel, and plot the central point.
e Draw a trend line of bus travel passing through the central point. Extend both the lines you have drawn as far as your scale allows.

4 a How much travel would you expect by rail in 1980?
b How much travel would you expect by bus and coach in 1980?
c In which year would you expect passenger-km travelled by rail to equal passenger-km by bus and coach? Why might this not be an accurate answer?

5 In 1973 private car travel was 360 billion passenger-km. Why do you think the 1975 figure was less?

6 Here is travel by two of the other methods of transport in 1973:
Rail, 35; bus, and coach, 54 billion passenger-km.
Plot these points on your graph (but don’t draw in new lines).

a How would this affect your answer in 4a for 1980 rail travel.
b How would it affect your answer for 1980 bus travel to question 4b?
c How might you change your answer to 4c? 

1 Possible answers include:
1 person travelled 18 km. 2 people travelled 9 km each.
3 people travelled 6 km each. 2 people travelled 5 km, and 1 person travelled 8 km.

2 a Bus and coach b Private cars; Air

Choice or Chance,    Retail Price Index,   Sampling the Census,  Smoking and Health, Testing Testing, Equal Pay

Level 1 Leisure for Pleasure,   Tidy Tables

Level 2 Fair Play

Level 3 Car Careers Multiplying People Cutting it Fine Phoney Figures

This unit is particularly relevant to: Social Sciences, General Knowledge, Mathematics, Economics, Commerce