Question 1 Chapter 12 page 13 of the ActEd notes states that, when carrying out hypothesis tests on the value of binomial parameter p or Poisson parameter lambda, we should adjust the value of X towards the mean of the distribution under H0. I understand the need to do a continuity correction, but am not clear on why we are always adjusting towards the mean rather than away from it. Can anyone explain this? Question 2 The ActEd notes do not appear to apply a continuity correction when we use a normal approximation to find a confidence interval for binomial parameter p or Poisson parameter lambda. Why is this? Question 3 The ActEd notes also do not appear to apply a continuity correction when we use a normal approximation to find a confidence interval or conduct a hypothesis test on the difference between two binomial parameters p or Poisson parameters lambda. Some older threads on here have said that this is because the continuity corrections cancel each other out in these circumstances. However I am still not very clear on this. Can anyone explain this in more detail, preferably with an example? Furthermore, if the continuity corrections cancel each other out, presumably they are both adjustments in the same direction. How can we be confident that this will be the case? Thanks in advance for any help provided.
It's all to do with the calculation of the p-value - which is the probability of obtaining a more extreme statistic than the one obtained. Since for a discrete distribution we'll include the statistic (ie P(X>=...) or P(X<= ...) - this means that the continuity correction will be towards the centre of the distribution (ie towards the mean) Yes the Core Reading (the bold part of the notes) formulae do not apply a continuity correction for CI for single sample Poisson and binomial. I always assumed that this was because the formulae themselves are approximations (as we use an estimate on the denominator instead of the true value) so a continuity correction to improve accuracy seems a little inappropriate! For the CIs it will be the same reason as above. For the hypothesis test - I hope you can see (using the answer to Q1) that the p-value of each individual result would mean that each result would be shifted ½ towards the centre. Since they are subtracted the net result would indeed be zero. Hope this helps clarify things a bit.
