In this file :
At the solution of problem number 11 on top of page 7. The author writes that $$p(x >200) =p(z>\frac{(200-0.5)-\mu}{\sigma})$$ Shouldn't the right answer be : $$p(x >200) =p(z>\frac{(200+0.5)-\mu}{\sigma})$$ I mean it is $200+0.5$ instead of $200-0.5$. It is not a typo because the author makes the same statement in the next question by putting $175+0.5$ instead of $175-0.5$.
Context : The problem here is one of approximation of binomial distribution $B(n,p)$ by the normal distribution $N(\mu=np,\sigma^2=np(1-p))$ and we have to take account of a continuity correction factor. My understanding is that this is done by approximating $p(x > n)$ as $p(x>n+0.5)$ (here the inequality $x>n$ is strict) this is to prevent the approximation from being too large than the real value. The author is subtracting $0.5$ instead of adding $0.5$, am I missing something thank you for your help!
1 Answer
You are correct, presuming that "more than $200$" does not include $200$.
With the correct continuity correction you suggest, the calculated probability would be about $0.0438$ rather than the solution sheet's $0.0521$. The actual binomial probability is closer to $0.0452$, And this might matter if using a $95\%$ level of confidence
Using R:
> 1 - pnorm(200-0.5, 900*0.2, sqrt(900*0.2*(1-0.2)))
[1] 0.05208128
> 1 - pnorm(200+0.5, 900*0.2, sqrt(900*0.2*(1-0.2)))
[1] 0.04378725
> 1 - pbinom(200, 900, 0.2)
[1] 0.04519563
