MAS360 Practical and Applied Statistics Project Report (April 2019) Title: Prediction for Annual Demand for Secondary-School Teachers in Fifteen Years' Time in England, U.K.

1. Prediction for Annual Demand for
Secondary-School Teachers in Fifteen Years’
Time in England, U.K.
160198065
03 April 2019
Summary
The aim of this study is to predict the demand for the state-funded secondary-school teachers
in England, U.K. in the next fifteen years, for instance, from 2018 to 2032. The data
has been gathered from the school workforce censuses, as well as the school, pupils and
their characteristics reports. The statistical methods included are exploratory data analysis,
statistical modelling, ETS (error, trend and seasonality) model for forecasting and model
predictions. It was found that the number of secondary-school teachers would need to achieve
approximately 300 000 in next fifteen years, in order to satisfy the demand for increased
student enrolment.
1

2. Contents
Summary 1
1 Introduction 3
2 Data 4
2.1 Data Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Description of the Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3 Methods and Analysis 5
3.1 Exploratory Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3.1.1 Summary Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3.1.2 Scatterplot Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.1.3 Timeplots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.2 Statistical Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.3 Forecasting Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.4 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4 Results 16
4.1 Linear Regression Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.2 Predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
5 Conclusion 20
6 General Discussion 20
7 References 22
8 Appendices 23
8.1 Data Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
8.2 R Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2

3. 1 Introduction
There is a consensus among educational researchers that the school teachers are significantly
related to not only the achievement and performance, but also attitudes and behaviours of
students. Consequently, it is believed that the role of a teacher is vital to the teaching-learning
process in the classroom. In another major study, it is reported that there is a significant
robust association between education and economic growth. Thus, an educated and well-
trained future generation is essential, in order to produce a skilled labour force, which in turn
should result in economic expansion. During the post-World War II period, an enormous
increase in the birth rate resulted in an increase in the school enrolment, but only a small
number of graduates decided to be involved in teaching, so many worried that there would not
be sufficient teachers for the new students (Arnold et al., 1993). Furthermore, many feared
that students could suffer owing to larger class sizes and would be taught by teachers with
lack of subject expertise, leading to reduced student achievement and motivation (Arnold et
al., 1993).
Over the last two decades, a large growing body of literature has investigated the types of
teachers, who are more likely to quit the profession, and the causes of turnover. Ingersoll (2011)
points out that the teacher turnover is strongly associated with not only the characteristics of
teachers, but also the teaching field. Teachers working in mathematics, science, and special
needs education fields are more likely to leave their jobs (Ingersoll, 2011). Therefore, teacher
scarcity has been an alarming unresolved global issue, which has attracted the attention
of educators, school leaders, parents, and the economic and political authorities of every
nation. According to the United Nations Educational, Scientific and Cultural Organisation
(UNESCO) Institute for Statistics (2016), the additional number of teachers required globally
at the secondary level in 2030 will be approximately 44.4 million, of which 16.7 million are
needed to guarantee that there are on average less than 25 students per teacher in a classroom.
Futhermore, 27.6 million of this number are required to substitute the teachers who leave
the profession (UNESCO Institute for Statistics, 2016). In addition to this, a recent news
article published on the BBC suggests that the schools in England are experiencing a serious
shortage of teachers, which means an exacerbating pupil-teacher ratio (Coughlan, 2018). For
instance, it increased from around fifteen students per teacher in 2010 to seventeen last year
(Coughlan, 2018). Hence, the prediction about the demand for secondary-school teachers is
important to ensure there are sufficient numbers to meet the future demand.
This report seeks to forecast the annual demand for the secondary-school teachers in England
in fifteen years’ time. Firstly, an exploratory data analysis will be carried out on various
relationships between the factors and pattern of the time series data. Secondly, a statistical
linear model, which fits the data, will be identified. Thirdly, time series forecasting analysis
will be performed to predict the values of the independent variables for the next fifteen years
(2018-2032). Fourthly, the predicted data will be used to forecast the annual demand for the
secondary-school teachers in England. The conclusion provide an overview and summary of
this research. A general discussion will be added at the end to complete this report, where
other research papers and journal articles will be considered, as well as to what extent the
results were successful.
3

4. 2 Data
year teachers mathteachers pupils leavers overcrowded
2001 193.2 .. 3248.605 .. ..
2002 196.7 .. 3280.250 .. ..
2003 203.2 .. 3327.750 .. ..
2004 206.9 .. 3353.100 .. ..
2005 217.4 .. 3348.950 .. ..
2006 219.2 .. 3347.035 .. 7.5
2007 220.9 .. 3325.330 .. 7.2
2008 221.5 .. 3294.250 .. 6.7
2009 222.4 .. 3277.805 .. 6.6
2010 219.0 33.0 3277.780 .. 6.5
2011 215.2 33.0 3261.785 9.40 6.6
2012 215.7 32.8 3233.940 8.80 6.5
2013 214.2 33.3 3209.055 9.30 6.2
2014 213.4 33.4 3180.175 9.80 5.8
2015 210.9 33.7 3183.280 10.2 5.9
2016 208.2 34.4 3191.780 10.3 6.5
2017 204.2 34.6 3221.575 10.4 7.4
2018 .. .. 3258.451 .. 8.0
Table 1: In the table showing the data used in the study, the following terms are explained:
“teachers” denotes the annual number of secondary-school teachers (in thousands); “math-
teachers” denotes the annual number of secondary-school Mathematics teachers; “pupils”
denotes the annual number of secondary-school pupils (in thousands); “leavers” denotes
the percentage of secondary-school teachers leaving the profession; “overcrowded” denotes
the percentage of secondary-school classes with more than thirty pupils; “..” denotes the
unavailable data.
2.1 Data Sources
The research analysis utilised in this study was cross-sectional analysis with the use of time
series data. The data for the number of secondary-school teachers for the period 2001-2004
comes from the school workforce census 2008, and from 2005 to 2017 comes from the school
workforce census 2017. The data for the number of secondary-school teachers will be used as
a measure of the demand for secondary-school teachers annually. The data for the number of
secondary-level Mathematics teachers (2010-2017) will be used as a measure of the demand
for secondary-school Mathematics teachers annually, and is sourced from the school workforce
census from 2010 to 2017. Besides that, the data for the number of secondary-school pupils for
the period of 18 years (2001-2018) is sourced from the school, pupils and their characteristics
4

5. national tables 2011 and 2018. Moreover, the data for the annual percentage of secondary-
school teachers leaving the profession (2011-2017) originates from the school workforce census
2017. Furthermore, the data for the annual percentage of secondary-school class with more
than thirty pupils is sourced from the school, pupils and their characteristics national table
2018. These data are published by the U.K. Department of Education, therefore they are
reliable, unbiased, valid and credible. However, some of the data are unavailable and due to
time limitation, only the data available will be used in the statistical analysis.
2.2 Description of the Data
Table 1 on page 5 shows the data used in this research. The data is gathered and recorded
in the Microsoft Excel before the analysis begin. There are five columns with titles: year,
teachers, pupils, leavers and overcrowded. The first column (year) states the year for the
observations. The second column (teachers) provides the annual number of secondary school
teachers. The third column (mathteachers) gives the annual number of secondary-school
Mathematics teachers. The proceding three columns (pupils, leavers and overcrowded)
present the annual number of secondary-school pupils, the percentage of secondary-school
teachers leaving the profession, and the percentage of secondary-school classes with over
thirty pupils. Both the numbers of secondary-school teachers and pupils are recorded in
thousand. The population of this study included state funded secondary schools in England
and all secondary-school full-time equivalent (FTE) teachers, who have been teaching in
secondary schools in England between 2001 and 2017. The percentage of secondary-school
teachers who leave the profession only include teachers with Qualified Teacher Status (QTS)
and unqualified teachers who leave the profession are not included. The leavers include those
who are out of service, retired and deceased. The percentage of secondary-school classes with
more than thirty pupils are computed by adding the percentage of those with 31-35 pupils
with the percentage ones with 36 or more pupils. The unavailable data is recorded as “..” in
Table 1.
3 Methods and Analysis
3.1 Exploratory Data Analysis
3.1.1 Summary Statistics
The exploratory data analysis (EDA) is a crucial process of conducting initial investigations on
the data, in order to detect patterns, discover anomalies and check assumptions in support of
summary statistics and graphical representations. Table 2 tabulates the summary statistics
for all the variables using the data available. The values for the number of observations, mean,
standard deviation, minimum, maximum, median, and interquartile range are calculated
using the commands in R. Some of the data is unavailable, so the number of observations
vary for each variable. The medians for the annual number of secondary school teachers and
5

6. pupils, as well as the percentage of secondary-school classes with more than thirty students,
are greater than the means, which implies that the data appear to be negatively skewed. On
the other hand, the means for the annual number of secondary-school Mathematics teachers,
and the percentage of the secondary-level teachers leaving the profession, are greater than
their medians, which indicates that the data may be positively skewed. The coefficients of
variation are computed using the following formula:
Variation coefficient =
Standard deviation
Mean
.
The coefficients of variation for all the variables are less than one, which are relatively low.
This means that the data points are spread out over a small range of values, in other words,
they are close to the mean values.
Variables teachers mathteachers pupils leavers overcrowded
Observations 17 8 18 7 13
Mean 211.8941 33.525 3267.828 9.742857 6.723077
Standard deviation 8.680544 0.6649382 57.35578 0.599603 0.6392102
Variation coefficient 0.040967 0.0198341 0.017552 0.061543 0.95077
Minimum 193.2 32.8 3180.175 8.8 5.8
Maximum 222.4 34.6 3353.1 10.4 8
Median 214.2 33.35 3269.783 9.8 6.6
Interquartile range 12.1 0.875 92.89375 0.9 0.7
Table 2: A table showing summary or descriptive statistics for all the variables using the
data available only, results from R.
3.1.2 Scatterplot Matrix
Figure 1 shows a scatterplot matrix of four variables, including response and independent,
with scatterplots below the diagonal, histogram and density plot of each variable on the
diagonal, as well as the correlation coefficients above the diagonal. The histogram for the
annual number of secondary-school teachers is positively skewed. Besides that, the histogram
for the number of secondary-school pupils depicts a normal distribution. Moreover, the bin
width of the histogram for the percentage of secondary-school teachers leaving the profession
is considerably small, owing to excessive individual data shown, and therefore the underlying
pattern of the data is hardly observed. The same goes for the histogram for the percentage
of secondary-school classes with over thirty pupils, but in this case, the density plot indicates
a normal distribution.
The scatterplot, which compares the numbers of secondary-level teachers and pupils, shows
that these two variables have an imperceptible positive correlation (r = 0.18). For instance, an
increase in the number of secondary-school students increases the number of secondary-school
6

7. teachers required. Besides that, the percentage of secondary-school teachers leaving the
profession and the demand for secondary-school teachers are negatively and substantially
associated (r = −0.87). This implies that the number of teachers increases as fewer teachers
leave the teaching profession. Furthermore, there is a positive relationship between the
percentage of secondary-school classes with more than thirty students, as well as the number
of secondary-school teachers needed (r = 0.12). This suggests that when the percentage of
secondary-school classes with no less than thirty pupils increases, the secondary-level teacher
demand would need to increase too.
pupils
9.09.510.0
3200 3250 3300 3350
195205215
9.0 9.5 10.0
−0.53
leavers
0.61
0.20
overcrowded
6.0 6.5 7.0 7.5 8.0
195 200 205 210 215 220
32003300
0.18
−0.87
6.06.57.07.58.0
0.12
teachers
Scatterplot matrix of four variables
Figure 1: A scatterplot matrix illustrates the relationships between the dependent and
independent variables, with scatterplots below the diagonal, histogram and density plot of
each variable on the diagonal, as well as the correlation coefficients above the diagonal. The
dependent variable is the annual number of secondary-school teachers, labelled as “teachers”.
The independent variables are: the annual number of secondary-school pupils, labelled
as “pupils”; the percentage of secondary-school teachers leaving the profession, labelled as
“leavers”; the proportion of secondary-school classes with more than thirty students, labelled
as “overcrowded”.
7

8. In addition, it is noteworthy to mention that some of the independent variables are correlated.
For example, the number of secondary-school pupils and the percentage of secondary-school
teachers leaving the profession are correlated (r = −0.53). Also, there is quite a high
correlation between the number of secondary-school pupils and the percentage of overcrowded
secondary-school classes (r = 0.61). Due to the fact that they are correlated, each of
them might directly or indirectly affect the demand of secondary-school teachers. If they
are statistically and economically relevant variables, the omission would result in a biased
estimator of β. On the other hand, if the variables are irrelevant, including them in the
regression analysis may also lead to bias. Therefore, this issue should be tackled carefully
when the statistical linear modelling is performed.
3.1.3 Timeplots
The timeplots, also referred to as time series graphs, are plotted for each variables. These
plots, which depict values of variables against year, are advantageous in the analysis section
because the patterns and trends of each variables could be easily identified. Figure 2
illustrates a timeplot of the annual number of secondary-school pupils from 2001 to 2018.
The number of secondary-school pupils increased until a peak was reached in 2004. According
to the Department for Education (2011), the upward trend was due to the increased rate of
birth in the late 1980s. Then, the number of secondary-level students declined gradually until
it reached a trough in 2014, because it was caused by the reduction in the rate of birth in the
1990s (Department for Education, 2011). The annual number of secondary-level pupils has
increases with the growth of 0.1 percent since January 2014 (Drake, 2015). In addition, Drake
(2015) mentioned that since January 2014, the proportion of minority ethnic students in
secondary schools has increased from 25.3 percent to 26.6 percent. For example, the number
of Asian secondary-school students has increased by more than 13 600 (Drake, 2015). The
percentage of overcrowded classes shown in Figure 5, could be explained by the number of
pupils, since they are correlated.
Figure 3 depicts a time series graph of the annual number of secondary-level teachers from
2001 to 2017. There was a rise in the secondary-level teacher number until 2009, after which
it declined. Inspecting the data frame, from 2008 to 2009, the number decline of secondary-
school teachers was approximately 9 000. A plausible explanation of the reduction could be
that from 2009 to 2011, the salaries of teachers were cut in some OECD nations, including
the U.K., Iceland, Hungary and Italy, which might have discouraged high-performing pupils
from a career in teaching (OECD, 2013). Moreover, it could be owing to the increased
percentage of secondary-teachers leaving the profession, as presented by the timeplot in
Figure 4. Examining the data published by the Department for Education (2017), the
number of secondary-school teachers leaving the profession, not due to retirement, increased
every year from 2011 to 2017. According to Ingersoll (2011), the association between the age
or teaching experience of teachers and their turnover rates could be explained by a U-shaped
curve. For instance, younger (no more than 30 year-olds) and older teachers (higher than
50 year-olds) are more likely to leave the profession, as compared to middle-aged teachers
(Ingersoll, 2011).
8

9. 3200325033003350
Annual number of secondary−school pupils 2001−2018
Year
Numberofsecondary−schoolpupils
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015
2016
2017
2018
Figure 2: A timeplot showing the annual number of secondary-school pupils (in thousands),
from 2001 to 2018.
195200205210215220
Annual number of secondary−school teachers 2001−2017
Year
Numberofsecondary−schoolteachers
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015
2016
2017
Figure 3: A timeplot showing the annual number of secondary-school teachers (in thousands),
from 2001 to 2017.
9

10. 9.09.510.0
Percentage of secondary−school teachers leaving the profession 2011−2017
Year
Pct.ofsecondary−schoolleavers
2011 2012 2013 2014 2015 2016 2017
Figure 4: A timeplot showing the annual percentage of secondary-school teachers leaving
the profession (in percent, %), from 2011 to 2017.
6.06.57.07.58.0
Percentage of secondary−school classes with more than thirty pupils 2006−2018
Year
Pct.ofclasseswithmorethan30pupils
2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018
Figure 5: A timeplot showing the annual percentage of secondary-school classes with more
than thirty pupils (in percent, %), from 2006 to 2018.
10

11. 3.2 Statistical Modelling
(a) Histogram of evals
evals
Density
−2 −1 0 1 2
0.00.10.20.30.4
−1.0 0.0 0.5 1.0
−1.5−0.50.51.01.5
(b) Normal Q−Q plot
Quantiles of standard normal
evals
Figure 6: (a) A histogram of the estimated standardised residuals (evals) against density
(Model 1). (b) A normal quantile-qiantile (Q-Q) plot of the evals (Model 1).
204 208 212 216
−1.5−0.50.51.01.5
(a) Residuals v.s. fitted plot
Fitted
Residuals
204 208 212 216
204206208210212214216
(b) Observed v.s. fitted plot
Fitted
Observed
Figure 7: (a) A scatterplot of residuals on the y-axis and fitted values on the x-axis (Model
1). (b) A scatterplot of observed values on the y-axis and fitted values on the x-axis (Model
1).
11

12. The variables, which are associated with the quantity demanded for new secondary-school
teachers annually, could be determined through the application of appropriate statistical
modelling. For year i, let Demandi be the annual demand for secondary-school teachers,
Pupilsi be the number of secondary-school pupils, Leaversi be the percentage of teachers
leaving the profession, Overcrowdedi be the percentage of classes with more than thirty
pupils, and εi be the error term, where all the error terms are normally distributed with a
mean of zero and a common variance, for instance, εi ∼ N(0, σ2
). The next phase is to check
the assumption εi ∼ N(0, σ2
) with the use of a histogram, a quantile-quantile (Q-Q) plot,
and the Kolmogorov-Smirnov goodness-of-fit test. The model fitted to the data is
Model 1: Demandi = β0 + β1Pupilsi + β2Leaversi + β3Overcrowdedi + εi.
Figure 6(a) and Figure 6(b) illustrates the histogram and Q-Q plot for the standardised
residuals. It is not clear whether the histogram represents a symmetric bell shape. Therefore,
it is not guaranteed that the error terms are evenly distributed around mean zero, or satisfy the
normality assumption. However, the Q-Q plot appears to show that the normality assumption
is satisfied because the points seem to fall about the straight line. The theoritical quantiles
are plotted on the x-axis, where the quantiles are of the standard normal distribution with
mean zero and standard deviation one. Furthermore, the Kolmogorov-Smirnov goodness-of-fit
test has not provided any evidence against the null hypothesis
H0 : e1, ..., en
i.i.d
∼ N(0, 1),
where e1, ..., en are the residuals and i.i.d stands for independent and identically distributed.
For instance, the p-value is 0.9048, which is considerable, thus the model assumptions are
appropriate. The anomalies or outliers, and extreme values are not detected in the scatterplots
of residuals and observed values against fitted values in Figure 7. This model is the most
adequate for the data available because the F-statistic and R2
statistic for this model are
273.1 and 0.9964, respectively. This implies that this model illustrates substantial variation
in the data, for instance, approximately 99.6 percent.
3.3 Forecasting Analysis
The technique used in the forecasting analysis is from the forecast package (Hyndman et
al., 2019) in R. The ETS modelling is employed, where E, T and S symbolise error type,
trend type, and season type. E takes either “A” (additive), “M” (multiplicative) or “Z”
(automatically); T and S take either “N” (none), “A”, “M” or “Z” (Hyndman et al., 2019).
From the EDA in Section 3.1.2, due to incomplete and unavailable data, it is difficult to
detect trend and seasonality in the available data. As a consequence, in this research, additive
error, additive trend, and no seasonality are assumed, and thus the AAN model is employed.
The objective is to estimate the demand for secondary-school teachers infifteen years’ time.
Instead of forecasting the number of secondary-school teachers in the next fifteen years, the
factors, which might affect the teacher number, are considered in the forecasting analysis. It
is believed this may improve the final results. The data for the number of secondary-school
12

13. pupils and the percentage of secondary-school overcrowded classes in 2018 are available,
therefore the prediction starts from 2019.
Prediction for the number of pupils (2019−2032)
Year
Numberofsecondary−schoolpupils
300031003200330034003500
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015
2016
2017
2018
2019
2020
2021
2022
2023
2024
2025
2026
2027
2028
2029
2030
2031
2032
Figure 8: A timeplot presenting the prediction for the number of secondary-school pupils
from 2019 to 2032, ETS (AAN) output.
Figures 8, 9 and 10 illustrate the prediction for the number of secondary-school pupils from
2019 to 2032, the percentage of secondary-school teachers leaving the profession from 2018
to 2032, and the proportion of secondary-school classes with more than thirty pupils from
2019 to 2032. The blue line shows the forecast observations, the blue shaded area shows
the 95 percent upper and lower confidence intervals, and the grey shaded area shows the 80
percent upper and lower confidence intervals. The forecasted values are recorded in Table
5. The forecasted values for the annual number of secondary-level pupils are chosen, rather
than the upper or lower estimates. The lower 80 percent estimates for the percentage of
secondary-school teachers leaving the profession and the percentage of the overcrowded classes
have been selected. The reason for these choices will be explained in the results (Section 4).
The predicted percentage of the number of secondary-school teachers leaving the profession
shows a negative from 2027 to 2032. It is believed that none or considerably fewer teachers
will leave the profession and more people will choose a teaching career in the future. With the
new predicted data points obtained, predict command in R is used to predict the demand
for secondary-school teachers for the corresponding year. The results are tabulated in the
last column of Table 5. It is predicted that the demand for secondary-school teachers will
be likely to increased in the next fifteen years.
13

14. Prediction for the percentage of leavers (2018−2032)
Year
Pct.ofsecondary−schoolleavers
−20−10010203040
2011
2012
2013
2014
2015
2016
2017
2018
2019
2020
2021
2022
2023
2024
2025
2026
2027
2028
2029
2030
2031
2032
Figure 9: A timeplot presenting the prediction for the percentage of secondary-school
teachers leaving the profession from 2018 to 2032, ETS (AAN) output.
Prediction for the percentage of overcrowded classes (2019−2032)
Year
Pct.ofclasseswith>30pupils
0102030
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015
2016
2017
2018
2019
2020
2021
2022
2023
2024
2025
2026
2027
2028
2029
2030
2031
2032
Figure 10: A timeplot presenting the prediction for the percentage of secondary-school
classes with more than thirty pupils from 2018 to 2032, ETS (AAN) output.
14

15. 3.4 Extensions
33.033.534.034.5
Annual number of secondary−school Mathematics teachers 2010−2017
Year
NumberofMathematicsteachers
2010 2011 2012 2013 2014 2015 2016 2017
Figure 10: A timeplot showing the annual number of secondary-school Mathematics teachers
(in thousand), from 2010 to 2017.
(a) Histogram of evals
evals
Density
−2 −1 0 1 2
0.00.10.20.30.4
−1.0 0.0 0.5 1.0
−1.5−1.0−0.50.00.51.01.5
(b) Normal Q−Q plot
Quantiles of standard normal
evals
Figure 11: (a) A histogram of the estimated standardised residuals (evals) against density
(Model 2). (b) A normal quantile-qiantile (Q-Q) plot of the evals is displayed (Model 2).
15

16. From 2010 to 2017, there was overall an increasing trend for the annual number of state-
funded secondary-school Mathematics teacher in England, as shown in Figure 10. To predict
the demand for secondary-school Mathematics teacher in England in particular, the similar
statistical modelling in Section 3.2 and forecasting analysis in Section 3.3 are performed.
The number of secondary-school Mathematics teacher is now the dependent variable. The
regression model fitted to the data is
Model 2: MathDemandi = β0 + β1Pupilsi + β2Leaversi + β3Overcrowdedi + εi,
where MathDemandi denotes the demand for secondary-school Mathematics teachers. Simi-
larly, the normality assumption is checked using a histogram, a Q-Q plot and the Kolmogorov-
Smirnov goodness-of-fit test. Figure 11 (a) and Figure 11 (b) shows histogram and
Q-Q plot for the standardised residuals, respectively. The histogram depicts a slightly
left-skewed bell shape. The normal Q-Q plot verifies the normality assumption. The result
for the Kolmogorov-Smirnov test is 0.3775, so there is no clear evidence of non-normality,
even though conclusive results may be problematic with a fairly small sample. This model
adequately explains the variation in the data because the R2
-statistic is 0.9763, which is
considerably high. Finally, the demand for secondary-level Mathematics teachers in England
in the next fifteen years is predicted using the predict command in R, using the new data
forecasted in Section 3.3.
4 Results
4.1 Linear Regression Model
Df Sum Sq Mean Sq F value Pr(>F)
Pupils 1 9.325 9.325 71.975 0.003
Leavers 1 75.722 75.722 584.435 0.0002
Overcrowded 1 21.092 21.092 162.794 0.001
Residuals 3 0.389 0.130
Table 3: ANOVA results for Model 1. The p-values for each explanatory variables are
recorded in the last column.
Table 3 shows the ANOVA results, obtained from R, for Model 1. Table 4 shows the
regression output with confidence internals in the parentheses for Model 1. All of the
variables, for instance, Pupilsi, Leaversi and Overcrowdedi, are significant at 1 percent
significance level with the p-values of 0.003, 0.0002 and 0.001, respectively. Besides that, the
16

17. parameter estimates are β0 = 66.513, β1 = 0.067, β2 = −3.330 and β3 = −5.991. Therefore,
he simple linear regression model fitted is as follows:
Demandi = 66.51 + 0.067Pupilsi − 3.33Leaversi + 5.991Overcrowdedi.
The standardised regression coefficient for the annual number of secondary-school pupils
is 0.067. This indicates that if the number of secondary-school students is increased by a
thousand, the additional number of teachers needed would need to be increased by 67 teachers.
Furthermore, the beta coefficient for the percentage of secondary-school teachers leaving
the profession is −3.33. For instance, when the percentage of secondary-level teachers, who
leave the profession, is increased by one percent, the number of secondary-school teachers is
decreased by approximately 3 330 teachers. In addition to this, the regression beta coefficient
for the percentage of secondary-school classes with more than thirty students is −5.991. This
implies that an additional number of secondary-school teachers of roughly 5 991 teachers is
required to reduce the proportion of overcrowded secondary-level classes by one percent.
Dependent variable:
Teachers
Pupils 0.067∗∗∗
(0.048, 0.086)
Leavers −3.330∗∗∗
(−4.125, −2.535)
Overcrowded −5.991∗∗∗
(−6.911, −5.070)
Constant 66.513
(3.719, 129.308)
Observations 7
R2
0.996
Adjusted R2
0.993
Residual Std. Error 0.360 (df = 3)
F Statistic 273.068∗∗∗
(df = 3; 3)
Note: ∗
p<0.1; ∗∗
p<0.05; ∗∗∗
p<0.01
Table 4: A table for regression analysis for Model 1, with the annual number of secondary-
school teachers as dependent variable. The maximum likelihood estimates (MLE), number of
observations, R2
statistic, adjusted R2
, residual standard error, degree of freedom (df) and
F-statistic are shown. The values in the parentheses are the confidence intervals.
17

18. Df Sum Sq Mean Sq F value Pr(>F)
Pupils 1 0.465 0.465 21.168 0.019
Leavers 1 1.860 1.860 84.762 0.003
Overcrowded 1 0.389 0.389 17.744 0.024
Residuals 3 0.066 0.022
Table 5: ANOVA results for Model 2. The p-values for each explanatory variables are
recorded in the last column.
Dependent variable:
MathTeachers
Pupils −0.011∗
(−0.019, −0.003)
Leavers 0.598∗∗
(0.271, 0.925)
Overcrowded 0.814∗∗
(0.435, 1.193)
Constant 58.704∗∗
(32.860, 84.548)
Observations 7
R2
0.976
Adjusted R2
0.953
Residual Std. Error 0.148 (df = 3)
F Statistic 41.225∗∗∗
(df = 3; 3)
Note: ∗
p<0.1; ∗∗
p<0.05; ∗∗∗
p<0.01
Table 6: A table for regression analysis for Model 2, with the annual number of secondary-
school Mathematics teachers as dependent variable. The maximum likelihood estimates
(MLE), number of observations, R2
statistic, adjusted R2
, residual standard error, degree of
freedom (df) and F-statistic are shown. The values in the parentheses are the confidence
intervals.
18

19. The ANOVA results for Model 2 is recorded in Table 5. It is clear that all the explanatory
variables are significant at five percent significance level. For instance, the p-values for Pupilsi,
Leaversi and Overcrowdedi are 0.019, 0.003 and 0.024, respectively. From the regression
results tabulated in Table 6, the values for β0, β1, β2 and β3 are 58.704, −0.011, 0.598
and 0.814, respectively. The model just describes the data, but it is not truly mechanistic.
Such a scenario could be because it is possible that the effect may be difficult to explain
in isolation. Additionally, it could be that the variable Leaversi, which is the percentage of
secondary-school teachers leaving the profession, is not adequate. The data for the percentage
of secondary-school Mathematics teachers leaving the profession is not available. If it is
accessible, the data for the secondary-school teachers leaving the profession will be substituted
as the percentage of secondary-school Mathematics teachers leaving the profession. This
might provide a more suitable and mechanistic model eventually. Therefore, the model fitted
to the data is
MathDemandi = 58.70 − 0.011Pupilsi + 0.598Leaversi + 0.814Overcrowdedi.
4.2 Predictions
year pupils leavers overcrowded teachers mathteachers
2018 3258.451 9.8762554 8.000000 204.89 34.44377
2019 3259.493 9.2899009 8.167275 205.91 34.21765
2020 3260.539 8.4809550 8.232424 208.28 33.77527
2021 3261.585 7.5139768 8.180916 211.88 33.14346
2022 3262.631 6.4153251 8.029875 216.52 32.35190
2023 3263.676 5.2003125 7.790805 222.06 31.41914
2024 3264.722 3.8794996 7.472009 228.44 30.35822
2025 3265.768 2.4608446 7.079825 235.59 29.17906
2026 3266.814 0.9506692 6.619294 243.44 27.88956
2027 3267.860 -0.645828 6.094544 251.97 26.49617
2028 3268.906 -2.324264 5.509038 261.14 25.00435
2029 3269.951 -4.080873 4.865734 270.91 23.41875
2030 3270.997 -5.912373 4.167197 281.27 21.74341
2031 3272.043 -7.815865 3.415678 292.18 19.98190
2032 3273.089 -9.788769 2.613174 303.63 18.13739
Table 7: A table tabulating the predicted values, from 2018 to 2032 (15 years), for the number
of secondary-school pupils (pupils), the percentage of teachers leaving the profession (leavers),
the proportion of secondary-school classes with more than thirty students (overcrowded), and
the number of secondary-school teachers (teachers), including upper and lower estimates. All
values are in thousand.
19

20. In Table 7, the annual number of state-funded secondary-school pupils in England is
forecasted to increase in the next fourteen years, with the data available for 2018. The ETS
model predicts that there will be an increment of the number of secondary-school students by
approximately 1 000 students annually. Besides that, the annual percentage of the secondary-
school teachers leaving the profession is expected to decrease in the future. Based on the
prediction, it will be decreasing in an increasing rate. Moreover, regarding the percentage of
the secondary-level classes with more than thirty students, it is predicted to decrease in an
increasing rate. Using the new data forecasted, the demand for secondary-school teachers
can be predicted using the predict command in R and recorded in Table 5. It is predicted
that demand for secondary-school teachers will increase in a increasing rate in next fifteen
years. Regarding the prediction for the demand for secondary-school Mathematics teachers,
as expected, the prediction is incorrect and misleading, because the model only describes the
data, but it is not mechanistic, as aforementioned.
5 Conclusion
To conclude, this paper finds that the annual number of secondary-school pupils, the percent-
age of secondary-school teachers leaving the profession, and the proportion of secondary-school
classes with more than thirty students, are significantly correlated with the annual num-
ber of secondary-school teachers. Using the ETS model for forecasting, the data for the
explanatory variables for the next fifteen years is then obtained. If is found that the annual
number of secondary-school pupils is expected to increase by around 1 000 students annually.
The percentage of secondary-school teachers leaving the profession and the proportion of
secondary-school overcrowded classes are estimated to be decreasing in an increasing rate.
Then, the annual demand for secondary-school teachers in England is predicted. It is found
that the demand for secondary-school teachers will increase, and in 2032, the predicted
demand for secondary-school teachers will be approximately 303 630.
6 General Discussion
As recorded in the Initial Teacher Training (ITT) census 2018-19, the number of trainee
teachers recruited at secondary level was approximately 3 300 (17%) below the targeted figure
(Foster, 2019). For instance, there was a shortfall of around 920 and 640 for Mathematics
and Physics teachers at secondary-level (Foster, 2019). As aforementioned, the forecasted
values for the annual number of secondary-level pupils were selected, because it is more likely
to be increasing in such a trend. According to the Department for Education (2018), in
2027, the overall population in the state-funded secondary schools is anticipated to reach 3
267 000. For instance, there should be approximately a 15 percent increment, or 418 000
more than it was in 2018. This figure matches the ETS model forecasting results in Table 4,
which is approximately 3 267 860 pupils in 2027. However, the Department for Education
(2018) also mentioned that it is vital to understand the overall influence of the uncertainty
20

21. in the forecasting, for example, the effects on net migration and fertility. It is believed that
smaller class size increases not only the effectiveness and flexibility of teaching, but also
the potential for pupils to learn more effectively. As a result, class size reduction has been
and will always be one of the main objectives of the Department for Education, in order to
increase effective learning and teaching in the classroom. Therefore, the secondary-school
class size is anticipated to decrease in coming years.
In March 2016, approximately 250 000 teachers, aged below sixty and formerly employed in
state-funded schools in England, left teaching profession (Foster, 2019). In 2017, the number
of teachers leaving the profession was greater than the number of teachers entering the
profession (Foster, 2019). Some reasons for teacher turnover are workload, school situation,
personal circumstances and low salary. Regarding workload, it is believed that teachers spend
substantial time on planning lessons, marking pupils’ works, and managing data, rather
than actul classroom teaching. In addition, some factors contribute to teachers leaving the
profession include lack of support, school policy and behaviour or attitudes of students. It is
forecasted that the rate of secondary-school teachers leaving the profession would decrease in
the future. In 2014, the Education Secretary, Nicky Morgan, started the ‘Workload Challenge’,
a survey of teachers intending to recognise the causes of immoderate workload and the ways
to reduce it (Foster, 2019). Therefore, the demand for secondary-school teachers in England
is expected to rise in the next fifteen years.
21

22. 7 References
• Arnold, C. L., Choy, S. P. and Bobbitt, S. A., (1993). Modeling teacher supply and
demand, with commentary. Washington, DC: National Center for Education Statistics.
• Coughlan, S., (2018). England’s schools face ‘severe’ teacher shortage. BBC News
[online]. 30 August. [Viewed 13 March 2019]. Available from: https://www.bbc.co.uk/
news/education-45341734.
• Department for Education., (2011). Class size and education in England evidence
report Research Report DFE-RR169 [online]. [Viewed 25 March 2019]. Available
from: https://assets.publishing.service.gov.uk/government/uploads/system/uploads/
attachment_data/file/183364/DFE-RR169.pdf.
• Department for Education., (2018). National pupil projections - Future trends in
pupil numbers: July 2018 [online]. London: Department for Education. Available
from: https://assets.publishing.service.gov.uk/government/uploads/system/uploads/
attachment_data/file/723851/2018Release_Projections_Text.pdf.
• Drake, R., (2015). Schools, pupils and their characteristics: January 2015 [on-
line]. London: Department for Education. [Viewed 25 March 2019]. Available
from: https://assets.publishing.service.gov.uk/government/uploads/system/uploads/
attachment_data/file/433680/SFR16_2015_Main_Text.pdf.
• Foster, D., (2019). Teacher recruitment and retention in England Briefing Paper
Number 7222 [online]. London: By the authority of the House of Commons. [Viewed 27
March 2019]. Available from: https://researchbriefings.parliament.uk/ResearchBriefing/
Summary/CBP-7222#fullreport.
• Hyndman, R. J., Athanasopoulos, G., Bergmeir, C., Caceres, G., Chhay, L., O’Hara-
Wild, M., Petropoulos, F., Razbash, S., Wang, E., Yasmeen, F., R Core Team, Ihaka,
R., Reid, D., Shaub, D., Tang, Y. and Zhou, Z., (2019). Package ‘forecast’ [online].
[Viewed 18 March 2019]. Available from: https://cran.r-project.org/web/packages/
forecast/forecast.pdf.
• Ingersoll, R. M., (2001). Teacher turnover and teacher shortages: An organizational
analysis. American Educational Research Journal [online]. 38(3), 499-534. [Viewed 13
March 2019]. Available from: doi: 10.3102/00028312038003499.
• OECD., (2013). What is the impact of the economic crisis on public education spending?.
Education Indicators in Focus, No. 18 [online]. [Viewed 26 March 2019]. Available from:
https://doi-org.sheffield.idm.oclc.org/10.1787/5jzbb2sprz20-en.
• UNESCO Institute for Statistics., (2016). The world needs almost 69 million new
teachers to reach the 2030 education goals [online]. Canada: UIS. UIS Fact Sheet
No. 39. [Viewed 13 March 2019]. Available from: https://unesdoc.unesco.org/ark:
/48223/pf0000246124.
22

23. 8 Appendices
8.1 Data Sources
• Department for Education., (2011). School workforce in England: November 2010 [on-
line]. London: Department for Education. [Viewed 12 March 2019] Available from: https:
//www.gov.uk/government/statistics/school-workforce-in-england-november-2010.
• Department for Education., (2012). School workforce in England: November 2011 [on-
line]. London: Department for Education. [Viewed 12 March 2019] Available from: https:
//www.gov.uk/government/statistics/school-workforce-in-england-november-2011.
• Department for Education., (2013). School workforce in England: November 2012 [on-
line]. London: Department for Education. [Viewed 12 March 2019] Available from: https:
//www.gov.uk/government/statistics/school-workforce-in-england-november-2012.
• Department for Education., (2014). School workforce in England: November 2013 [on-
line]. London: Department for Education. [Viewed 12 March 2019] Available from: https:
//www.gov.uk/government/statistics/school-workforce-in-england-november-2013.
• Department for Education., (2015). School workforce in England: November 2014 [on-
line]. London: Department for Education. [Viewed 12 March 2019] Available from: https:
//www.gov.uk/government/statistics/school-workforce-in-england-november-2014.
• Department for Education., (2016). School workforce in England: November 2015 [on-
line]. London: Department for Education. [Viewed 12 March 2019] Available from: https:
//www.gov.uk/government/statistics/school-workforce-in-england-november-2015.
• Department for Education., (2017). School workforce in England: November 2016 [on-
line]. London: Department for Education. [Viewed 12 March 2019] Available from: https:
//www.gov.uk/government/statistics/school-workforce-in-england-november-2016.
• Department for Education., (2018). School workforce in England: November 2017 [on-
line]. London: Department for Education. [Viewed 12 March 2019] Available from: https:
//www.gov.uk/government/statistics/school-workforce-in-england-november-2017.
• Department for Education., (2018). Schools, pupils and their characteristics: January
2018. London: Department for Education. [Viewed 12 March 2019] Available from:
https://www.gov.uk/government/statistics/schools-pupils-and-their-characteristics-
january-2018.
23

24. 8.2 R Codes
par(pty="m",mai=c(0.85,0.85,0.7,1),cex.lab=0.9,cex.axis=0.9)
variables<-workforce_copy_3[,c(3,5,8,2)]
library(psych)
pairs.panels(variables,smooth=TRUE,scale=FALSE,density=TRUE,pch=18,
method="pearson",hist.col="lightskyblue",
main="Scatterplot matrix of four variables",
ellipses=TRUE)
Pupils<-c(3248.605,3280.25,3327.75,3353.1,3348.95,3347.035,3325.33,
3294.25,3277.805,3277.78,3261.785,3233.94,3209.055,3180.175,
3183.28,3191.78,3221.575,3258.451)
Pupils.year<-c(2001:2018)
par(pty="m",mai=c(0.85,0.85,0.7,1),cex=.85,cex.lab=1,cex.axis=1)
plot(Pupils.year,Pupils,xaxt="n",pch=1,col="dodgerblue4",type="o",
main="Annual number of secondary-school pupils 2001-2018",
xlab="Year",ylab="Number of secondary-school pupils")
axis(1,at=seq(2001,2018,by=1),las=2)
Teachers<-c(193.2,196.7,203.2,206.9,217.4,219.2,220.9,221.5,222.4,219,
215.2,215.7,214.2,213.4,210.9,208.2,204.2)
Teachers.year<-c(2001:2017)
par(pty="m",mai=c(0.85,0.85,0.7,1),cex=.85,cex.lab=1,cex.axis=1)
plot(Teachers.year,Teachers,xaxt="n",pch=1,col="dodgerblue4",type="o",
main="Annual number of secondary-school teachers 2001-2017",
xlab="Year",ylab="Number of secondary-school teachers")
axis(1,at=seq(2001,2017,by=1),las=2)
Leavers<-c(9.4,8.8,9.3,9.8,10.2,10.3,10.4)
Leavers.year<-c(2011:2017)
par(pty="m",mai=c(0.85,0.85,0.7,1),cex=.85,cex.lab=1,cex.axis=1)
plot(Leavers.year,Leavers,xaxt="n",pch=1,col="dodgerblue4",type="o",
main="Percentage of secondary-school teachers leaving the profession
2011-2017",
xlab="Year",ylab="Pct. of secondary-school leavers")
axis(1,at=seq(2011,2017,by=1),las=1)
Overcrowded<-c(7.5,7.2,6.7,6.6,6.5,6.6,6.5,6.2,5.8,5.9,6.5,7.4,8)
Overcrowded.year<-c(2006:2018)
par(pty="m",mai=c(0.85,0.85,0.7,1),cex=.85,cex.lab=1,cex.axis=1)
plot(Overcrowded.year,Overcrowded,xaxt="n",pch=1,col="dodgerblue4",
main="Percentage of secondary-school classes with more than thirty
pupils 2006-2018",
type="o",xlab="Year",ylab="Pct. of classes with more than 30 pupils")
24

25. axis(1,at=seq(2006,2018,by=1),las=1)
lm1<-lm(teachers~overcrowded+leavers+pupils)
library(MASS)
par(mfrow=c(1,2))
evals<-stdres(lm1)
par(pty="m",mai=c(0.85,0.85,0.7,1),cex=.85,cex.lab=1,cex.axis=1)
hist(evals,nclass=4,col="lightskyblue",freq=F,
main="(a) Histogram of evals")
par(pty="m",mai=c(0.85,0.85,0.7,1),cex=.85,cex.lab=1,cex.axis=1)
qqnorm(evals,main="(b) Normal Q-Q plot",xlab="Quantiles of standard normal",
ylab="evals",pch=1,col="dodgerblue3")
abline(0,1,col="firebrick3",lwd=2,lty=1)
lm1<-lm(teachers~pupils+leavers+overcrowded,y=TRUE)
par(mfrow=c(1,2))
par(pty="m",mai=c(0.85,0.85,0.7,1),cex=.85,cex.lab=1,cex.axis=1)
plot(fitted(lm1),evals,xlab="Fitted",ylab="Residuals",
axes=T,ylim=c(),xlim=c(),pch=1,col="dodgerblue4",
main="(a) Residuals v.s. fitted plot")
grid(lwd=2)
par(pty="m",mai=c(0.85,0.85,0.7,1),cex=.85,cex.lab=1,cex.axis=1)
plot(fitted(lm1),lm1$y,xlab="Fitted",ylab="Observed",
axes=T,ylim=c(),xlim=c(),pch=1,col="dodgerblue4",
main="(b) Observed v.s. fitted plot")
abline(a=0,b=1,col="firebrick3",lwd=2,lty=1)
grid(lwd=2)
library(forecast)
par(pty="m",mai=c(0.85,0.85,0.7,1),cex=.85,cex.lab=1,cex.axis=1)
Pupils.ts<-ts(Pupils,start=2001,end=2018,frequency=1)
Pupils.ets<-ets(Pupils.ts,model="AAN")
Pupils.forecast<-forecast(Pupils.ets,h=14)
plot(Pupils.forecast,xaxt="n",col="dodgerblue3",pch=1,type="o",
main="Prediction for the number of pupils (2019-2032)",
xlab="Year",ylab="Number of secondary-school pupils")
axis(1,at=seq(2001,2032,by=1),las=2)
library(forecast)
par(pty="m",mai=c(0.85,0.85,0.7,1),cex=.85,cex.lab=1,cex.axis=1)
Leavers.ts<-ts(Leavers,start=2011,end=2017,frequency=1)
Leavers.ets<-ets(Leavers.ts,model="AAN")
Leavers.forecast<-forecast(Leavers.ets,h=15)
plot(Leavers.forecast,xaxt="n",col="dodgerblue3",pch=1,type="o",
main="Prediction for the percentage of leavers (2018-2032)",
25

26. xlab="Year",ylab="Pct. of secondary-school leavers")
axis(1,at=seq(2001,2032,by=1),las=2)
library(forecast)
par(pty="m",mai=c(0.85,0.85,0.7,1),cex=.85,cex.lab=1,cex.axis=1)
Overcrowded.ts<-ts(Overcrowded,start=2006,end=2018,frequency=1)
Overcrowded.ets<-ets(Overcrowded.ts,model="AAN")
Overcrowded.forecast<-forecast(Overcrowded.ets,h=14)
plot(Overcrowded.forecast,xaxt="n",col="dodgerblue3",pch=1,type="o",
main="Prediction for the percentage of overcrowded classes (2019-2032)",
xlab="Year",ylab="Pct. of classes with > 30 pupils")
axis(1,at=seq(2001,2032,by=1),las=2)
Mathteachers<-c(33,33,32.8,33.3,33.4,33.7,34.4,34.6)
Mathteachers.year<-c(2010:2017)
par(pty="m",mai=c(0.85,0.85,0.7,1),cex=.85,cex.lab=1,cex.axis=1)
plot(Mathteachers.year,Mathteachers,xaxt="n",pch=1,col="dodgerblue4",
type="o",main="Annual number of secondary-school Mathematics
teachers 2010-2017",
xlab="Year",ylab="Number of Mathematics teachers")
axis(1,at=seq(2010,2017,by=1),las=1)
lm2<-lm(mathteachers~pupils+leavers+overcrowded,y=TRUE)
library(MASS)
par(mfrow=c(1,2))
evals.lm2<-stdres(lm2)
par(pty="m",mai=c(0.85,0.85,0.7,1),cex=.85,cex.lab=1,cex.axis=1)
hist(evals.lm2,nclass=4,col="lightskyblue",freq=F,
main="(a) Histogram of evals",xlab="evals")
par(pty="m",mai=c(0.85,0.85,0.7,1),cex=.85,cex.lab=1,cex.axis=1)
qqnorm(evals.lm2,main="(b) Normal Q-Q plot",
xlab="Quantiles of standard normal",ylab="evals")
abline(0,1,col="firebrick3",lwd=2,lty=1)
newdata.a<-data.frame(pupils=3258.451,leavers=9.8762554,
overcrowded=8.000000)
predict(lm1,newdata.a,interval="predict")
predict(lm2,newdata.a,interval="predict")
newdata.b<-data.frame(pupils=3259.493,leavers=9.2899009,
overcrowded=8.167275)
predict(lm1,newdata.b,interval="predict")
predict(lm2,newdata.b,interval="predict")
newdata.c<-data.frame(pupils=3260.539,leavers=8.4809550,
26

27. overcrowded=8.232424)
predict(lm1,newdata.c,interval="predict")
predict(lm2,newdata.c,interval="predict")
newdata.d<-data.frame(pupils=3261.585,leavers=7.5139768,
overcrowded=8.180916)
predict(lm1,newdata.d,interval="predict")
predict(lm2,newdata.d,interval="predict")
newdata.e<-data.frame(pupils=3262.631,leavers=6.4153251,
overcrowded=8.029875)
predict(lm1,newdata.e,interval="predict")
predict(lm2,newdata.e,interval="predict")
newdata.f<-data.frame(pupils=3263.676,leavers=5.2003125,
overcrowded=7.790805)
predict(lm1,newdata.f,interval="predict")
predict(lm2,newdata.f,interval="predict")
newdata.g<-data.frame(pupils=3264.722,leavers=3.8794996,
overcrowded=7.472009)
predict(lm1,newdata.g,interval="predict")
predict(lm2,newdata.g,interval="predict")
newdata.h<-data.frame(pupils=3265.768,leavers=2.4608446,
overcrowded=7.079825)
predict(lm1,newdata.h,interval="predict")
predict(lm2,newdata.h,interval="predict")
newdata.i<-data.frame(pupils=3266.814,leavers=0.9506692,
overcrowded=6.619294)
predict(lm1,newdata.i,interval="predict")
predict(lm2,newdata.i,interval="predict")
newdata.j<-data.frame(pupils=3267.860,leavers=-0.645828,
overcrowded=6.094544)
predict(lm1,newdata.j,interval="predict")
predict(lm2,newdata.j,interval="predict")
newdata.k<-data.frame(pupils=3268.906,leavers=-2.324264,
overcrowded=5.509038)
predict(lm1,newdata.k,interval="predict")
predict(lm2,newdata.k,interval="predict")
27

28. newdata.l<-data.frame(pupils=3269.951,leavers=-4.080873,
overcrowded=4.865734)
predict(lm1,newdata.l,interval="predict")
predict(lm2,newdata.l,interval="predict")
newdata.m<-data.frame(pupils=3270.997,leavers=-5.912373,
overcrowded=4.167197)
predict(lm1,newdata.m,interval="predict")
predict(lm2,newdata.m,interval="predict")
newdata.n<-data.frame(pupils=3272.043,leavers=-7.815865,
overcrowded=3.415678)
predict(lm1,newdata.n,interval="predict")
predict(lm2,newdata.n,interval="predict")
newdata.o<-data.frame(pupils=3273.089,leavers=-9.788769,
overcrowded=2.613174)
predict(lm1,newdata.o,interval="predict")
predict(lm2,newdata.o,interval="predict")
28