Misconception Mutt extract from chapter 2
The Misconception Mutt was dragging his owner down the street one day. His owner thought that he was sniffing lampposts for interesting smells, but the mutt was distracted by thoughts of confidence intervals.
‘A 95% confidence interval has a 95% probability of containing the population parameter value,’ he wheezed as he pulled on his lead.
A ginger cat emerged. The owner dismissed his perception that the cat had emerged from a solid brick wall. His dog pulled towards the cat in a stand-off. The owner started to check his text messages.
‘You again?’ the mutt growled.
The cat considered the dog’s reins and paced around, smugly displaying his freedom. ‘I’m afraid you will see very much more of me if you continue to voice your statistical misconceptions,’ he said. ‘They call me the Correcting Cat for a reason’.
The dog raised his eyebrows, inviting the feline to elaborate.
‘You can’t make probability statements about confidence intervals,’ the cat announced.
‘Huh?’ said the mutt.
‘You said that a 95% confidence interval has a 95% probability of containing the population parameter. It is a common mistake, but this is not true. The 95% reflects a long-run probability.’
The cat raised his eyes to the sky. ‘It means that if you take repeated samples and construct confidence intervals, then 95% of them will contain the population value. That is not the same as a particular confidence interval for a specific sample having a 95% probability of containing the value. In fact, for a specific confidence interval, the probability that it contains the population value is either 0 (it does not contain it) or 1 (it does contain it). You have no way of knowing which it is.’ The cat looked pleased with himself.
‘What’s the point of that?’ the dog asked.
The cat pondered the question. ‘It is important if you want to control error,’ he eventually answered. ‘If you assume that the confidence interval contains the population value then you will be wrong only 5% of the time if you use a 95% confidence interval.’
The dog sensed an opportunity to annoy the cat. ‘I’d rather know how likely it is that the interval contains the population value,’ he said.
‘In which case, you need to become a Bayesian,’ the cat said, disappearing indignantly into the brick wall.
The mutt availed himself of the wall, hoping it might seal the cat in for ever.