Solution to question 4.4
The first step in calculating the 95% confidence interval is calculate the standard error. We use the formula:
Applying the formula in turn to the three sets of figures gives us the standard errors in the table below.
| Material | Number of specimens | Square root of number of specimens | Standard deviation of fluoride release | Standard error |
|---|---|---|---|---|
| 5 | 2·24 | 4·85 | 2·17 | |
| Chem-Fil Superior | 5 | 2·24 | 2·55 | 1·14 |
| Compoglass | 5 | 2·24 | 1·02 | 0·46 |
(I have reported my intermediate results above to two decimal places of accuracy. In fact, I did not round these up whilst I was doing my calculations, I kept them as accurately as my calculator allowed and only rounded up at the final stage.)
Now, the 95% confidence interval is the interval:
From 'sample mean - 1·96 standard errors' to 'sample mean + 1·96 standard errors'
So we calculate 1·96 standard errors for each group of patients and subtract and add this to their respective means.
| Material | Mean fluoride release (mg mm-2) | 1·96 standard errors | Mean - 1·96 standard errors | Mean + 1·96 standard errors |
|---|---|---|---|---|
| Ketac-Fil | 30·62 | 5·27 | 25·35 | 35·89 |
| Chem-Fil Superior | 12·69 | 2·23 | 10·45 | 14·92 |
| Compoglass | 10·35 | 0·89 | 9·46 | 11·24 |
So we would report our results as follows:
- Mean fluoride release from Ketac-Fil is 30·62mg mm-2 (95% CI from 25·35 to 35·89)
- Mean fluoride release from Chem-Fil Superior is 12·69mg mm-2 (95% CI from 10·45 to 14·92)
- Mean fluoride release from Compoglass is 10·35mg mm-2 (95% CI from 9·46 to 11·24)
Note that I have put the units after the mean. This is compulsory; without units, numbers are clinically meaningless.
The 95% confidence intervals for Chem-Fil Superior and Compoglass overlap by a considerable amount. For example, a fluoride leakage of 11mg mm-2 is a plausible value for the means of both materials. We are unable to tell if there is a real difference between the two materials. By contrast none of the range values in the confidence interval for Ketac-Fil would be a plausible value for the mean leakage of either of the other two materials. We might conclude that there is a real difference between Ketac-Fil and the other two materials
This method of looking at the confidence intevals of the means of different samples is a good first step to seeing if a real difference is likely to exist between them. There is a more formal and correct approach: statistically testing for a difference and calculating a confidence interval for the difference. We shall be learning how to do this in a later unit.
