Australia Post claims to deliver 96% of letters on time but only 52 out of 59 letters arrived on time in a study.

X   ~   binomial (n = 59, π)

H0:   π = 0.96        HA:   π < 0.96

A simulation was used earlier to find the approximate p-value for this test by generating repeated samples from a population with π = 0.80. This page illustrates that the binomial distribution can give the p-value directly without simulation.

The p-value is a sum of binomial probabilities

Firstly repeat the simulation — click Accumulate then hold down Simulate until about 100 samples of 59 letters have been generated. This gives an approximation to the p-value.

Now select Binomial distribution from the pop-up menu. The exact probability of getting 52 or fewer weapons detected is 0.009 if π = 0.96. This is the p-value for the test.

The conclusion is the same as before:

Getting only 52 letters delivered on time would be unlikely if Australia Post's claim was correct so there is strong evidence that fewer than 96% are delivered on time.


The Herald-Sun newspaper published the following article on November 25 1992.

Doubt has been cast over Australia Post's claim of delivering 96 per cent of standard letters on time.

A survey conducted by the Herald-Sun in Melbourne revealed that less than 90 per cent of letters were delivered according to the schedule.

Herald-Sun staff posted 59 letters before the advertised...

Campbell Fuller, Herald-Sun, 25 November 1992.

Is the author justified in disputing Australia Post's claim that 96% of letters are delivered on time?