Course Notes - Department of Mathematics and Statistics

WHEN IS THE USE OF A NORMAL APPROXIMATION APPROPRIATE?A variety of studies suggest that 11% of the world population isleft-handed (for interest, 34 out of 304, or 11.2% of STAT110 studentswho gave answers for the questionnaire are left-handed). Consider thesituation where the sample size is two.Sample size = 2Here the mean is n × π = 2 × 0.11 = 0.22 and the standarddeviation is √ n × π × (1 − π) = √ 2 × 0.11 × 0.89 = 0.445. nπ ±3 √ nπ(1 − π) = 0.22 ± 3 × 0.445 = -1.11 and 1.55. These values arenot between 0 and n = 2.• For this example n is small and π is is far from 0.5. The distributionis not symmetrical.• nπ ± 3 √ nπ(1 − π) does not give two values between 0 and n.Unappropriate to use the normal approximation to the binomial inthis situation.82