# Exercises: Sample size calculations

##### Question 11.1

Two studies were carried out to compare the mean plaque index (0-10) of people using two brands of toothpaste (*Fluoriwonder* and *Supercheapo*). In study **A** each group had 10 people in it, in study **B** each group had 100 people in it. The results of the two studies are given below.

Study | Difference in mean plaque index (Fluoriwonder - Supercheapo) | 95% Confidence Interval | P |
---|---|---|---|

A | 3·1 | -3·5 to 8·7 | 0·131 |

B | 3·0 | 1·3 to 4·7 | 0·001 |

How would you interpret these results using the concepts you have learned in the lecture?

##### Question 11.2

We wish to carry out a similar study to the one above to compare the mean plaque index of people using Fluoriwonder and Fairlycheapo toothpastes. From pilot studies we think that the standard deviation will be 3·0 units. The difference worth finding is 1·5 units. If we want the study to have a level of significance of 5% and a power of 90%, how many patients do we need?

##### Question 11.3

Find the number of patients required in the study outlined in question 2 under the following conditions:

Standard deviation | Difference worth finding | Power | Significance | |
---|---|---|---|---|

a) | 8·0 | 3·0 | 90% | 5% |

b) | 6·0 | 2·5 | 90% | 5% |

c) | 6·0 | 3·0 | 80% | 5% |

d) | 6·0 | 3·0 | 90% | 1% |

##### Question 11.4

Once again we wish to carry out a study to compare Fluoriwonder and Fairlycheapo toothpastes. Pilot data indicates that 65% subjects using Fairlycheapo toothpaste over a period of 1 month showed an improvement of at least 2 units in their plaque scores. We consider that the extra cost of Fluoriwonder toothpaste would be justified if 80% of subjects show an improvement of at least 2 units. How many patients do we need to get 80% power for this study at a significance level of 5%?

If the pilot data had shown that 45% of Farlycheapo users improved and we felt that we would recommend Fluoriwonder if 60% of users improved, how many patients would we need?

##### Question 11.5

Before a study comparing the mean plaque index of people using Fluoriwonder and Fairlycheapo toothpastes we think that: the standard deviation of the plaque indices will be around 5 units; the difference worth finding is 4·0 units; the power should be 80%; and the level of significance should be 1%.

Write a few sentences to put in the methods section of a research proposal giving the sample size.