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"It has long been an axiom of mine that the little things are infinitely the most important" (Sherlock Holmes)

# Confidence Interval of a Mean Normal approximation method  ### Large sample from an 'infinite' population

#### Worked example

The mean haemoglobin content for a random sample of 225 school children is 108 g/l with a standard deviation (s) of 41 g/l. The 95% confidence interval is given by: 95% CI( ) = 108 ± 1.96 × 41 √225 = 102.6 to 113.4 Hence the range 102.6 to 113.4 g/L should enclose the true mean haemoglobin content on 95% of occasions it is calculated, assuming that range is calculated in a similar fashion from randomly selected samples.

### Small sample from an 'infinite' population

#### Worked example

The mean GBH (girth at breast height) for a random sample of 10 aspen trees is 34.7 cm with a standard deviation (s) of 6.77 cm. A histogram of the frequency distribution suggests that it may be close to normal.

Remember: you will always have less confidence in statistics derived from small samples as they may not be representative of the population.

Bearing this in mind....

The 95% confidence interval is given by: 95% CI( ) = 34.7 ± 2.26 × 6.77 √10 = 29.9 to 39.5 Hence the range 29.9 to 39.5 cm should enclose the true mean girth at breast height on 95% of occasions it is calculated, assuming that range is calculated in a similar fashion from randomly selected samples.

### Sample from a small finite population

#### Worked example

The mean serum copper content for a random sample of 74 white rhinos out of the total world population of 560 white rhinos is 6.3 μmol/L, with a standard deviation of 2.3 μmol/L. The 95% confidence interval is given by: 95% CI ( ) = 6.3 ± 1.96 × 2.3 × √( 1 − 74 )  √74 560 = 5.81 to 6.79 Hence the range 5.81 to 6.79 μmol/L should enclose the true mean serum copper content on 95% of occasions it is calculated, assuming that range is calculated in a similar fashion from randomly selected samples.

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