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There is a concern expressed that the general dental practitioners are more willing to work for NHS (National Health Service) after qualification however, later they choose private practice after gaining professional experience. To verify this concern, data regarding how many years the dentists have been practicing since qualification was collected for three sample groups of dentists: 1^{st}: The Dentists working in National Health Service only, 2^{nd}: The Dentists working in private practice only and 3^{rd}: the Dentists working in National Health Service and Private practice.
Thus Null hypothesis H0: The mean of the different groups will be equal
Alternative Hypothesis H1: The mean of the different groups will not be equal
The data collected is given below
NHS only 
Private only 
NHS and Private 
12 
15 
5 
27 
26 
16 
1 
38 
38 
26 
7 
24 
4 
33 
4 
6 
25 
42 
18 
12 
13 
23 
42 
22 
3 
31 
2 
10 
22 
28 
2 
16 
32 
3 
37 
37 
1
The scatter plot diagram of the above data is given below where 1 is the sample of dentists working in NHS only, 2 is the sample of dentists working in private practice only and 3 is the sample of dentists working in NHS and private practice. Each group has a sample of 12 dentists and the number of years they have practiced since qualification is noted. The each sample group is plotted on the x axis while the number of years they have practiced since qualification is on the y axis.
2
To find the one way analysis of variance
We know for the calculation of one way ANOVA, we need to calculate for each group mean and ∑ (X  X0)^2, where X is the sample data and X0 is the mean of the sample.
Thus, calculating the values for each group we have,
NHS only 
Private only 
NHS and Private 

Data 
(X  X0)^2 
Data 
(X  X0)^2 
Data 
(X  X0)^2 

12 
0.56 
15 
106.78 
5 
286.17 

27 
248.06 
26 
0.44 
16 
35.01 

1 
105.06 
38 
160.44 
38 
258.67 

26 
217.56 
7 
336.11 
24 
4.34 

4 
52.56 
33 
58.78 
4 
321.01 

6 
27.56 
25 
0.11 
42 
403.34 

18 
45.56 
12 
177.78 
13 
79.51 

23 
138.06 
42 
277.78 
22 
0.01 

3 
68.06 
31 
32.11 
2 
396.67 

10 
1.56 
22 
11.11 
28 
37.01 

2 
85.56 
16 
87.11 
32 
101.67 

3 
68.06 
37 
136.11 
37 
227.51 

Mean 
11.25 
25.33333 
21.91667 

∑(X  X0)^2 
1058.25 
1384.667 
2150.917 
We need to calculate mean and ∑ (X  X0) ^2 for overall data as well.
Overall data 
(X  X0)^2 
12 
56.25 
27 
56.25 
1 
342.25 
26 
42.25 
4 
240.25 
6 
182.25 
18 
2.25 
23 
12.25 
3 
272.25 
10 
90.25 
2 
306.25 
3 
272.25 
15 
20.25 
26 
42.25 
38 
342.25 
7 
156.25 
33 
182.25 
25 
30.25 
12 
56.25 
42 
506.25 
31 
132.25 
22 
6.25 
16 
12.25 
37 
306.25 
5 
210.25 
16 
12.25 
38 
342.25 
24 
20.25 
4 
240.25 
42 
506.25 
13 
42.25 
22 
6.25 
2 
306.25 
28 
72.25 
32 
156.25 
37 
306.25 
Mean = 19.5 

∑(X  X0)^2 = 5889 
Thus summarizing the overall data we have
NHS only 
Private only 
NHS and Private 
Overall 

Mean 
11.25 
25.33333 
21.91667 
19.5 
n 
12 
12 
12 
36 
Sum X 
135 
304 
263 
702 
∑(XX0)^2 
1058.25 
1384.667 
2150.917 
5889 
Thus difference within the group = 1058.25 + 1384.67 + 2150.91 = 4593.83
And the difference between the groups = 5889 – 4593.83 = 1295.17
The degree of freedom df for between group = 3 1 = 2, the degree of freedom df for within groups = (12 1) + (12 1) + (12 1) = 33. Thus, total degree of freedom = 33 + 2 =35
Mean is calculated by dividing sum of squares by df. (Laerd Statistics , 2013)
F = mean between groups/ mean within groups
Sum of squares 
df 
Mean 
F 

Between Group 
1295.17 
2 
647.59 
4.65 
Within Group 
4593.83 
33 
139.21 

Total 
5889 
35 
From F table for 95% confidence level we have, F critical = 3.284
F > F critical. Thus we reject the null hypothesis.
Thus we can say that there exist significant differences in the mean of the three groups. (Roberts, 2008)
c)
From the analysis done above we can conclude that the mean of the three samples which are 1^{st}: The Dentists working in National Health Service only, 2^{nd}: The Dentists working in private practice only and 3^{rd}: the Dentists working in National Health Service and Private practice are significantly different i.e. the number of years practiced by the dentist in the three samples is different and we can say that number of years of practice by dentists in National Health Service and private are different. (Laerd Statistics , 2013)
d)
There is a significant difference among the mean of the number of years of practice by dentists in National Health Service and private. By looking at the mean values of the number of years of practice by dentists, it can be said that the number of years of practice by dentists in private is more than the number of years of practice by dentists in NHS as the mean value of the sample for private only is more than two times the mean vale for NHS only. Thus we can conclude that the dentist choose private practice after gaining professional experience.
Penny, H 2006, Analysis of Variance, viewed 17 August 2016, <https://www.fil.ion.ucl.ac.uk/~wpenny/publications/spmbook/anova.pdf>
Roberts, M 2008, Analysis of Variance, Routledge.
Laerd Statistics 2013, Oneway ANOVA, viewed 17 August 2016, <https://statistics.laerd.com/statisticalguides/onewayanovastatisticalguide.php>
The Pennsylvania State University 2016, Five Step Hypothesis Testing Procedure, viewed 17 August 2016, <https://onlinecourses.science.psu.edu/stat200/node/67>
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