Contradiction between me and my professor solution in normal app. to binomial

Solution 1:

When using continuity correction (see the link below) when estimating $P(X \leq 40)$ where $X\sim \text{Binomial}(n,p)$ , you use the probability $P(Y \leq 40.5)$ where $Y \sim N(np,\sqrt{np(1-p)})$. Hence for estimating $P(X > 40)$ you would naturally have to use $P(Y>40.5)$. So this would mean your approach is correct.

However, your professor might be taking into account that for certain values of $p$ and $n$, the Normal approximation underestimates the Binomial distribution and hence choosing to correct for this by looking at $P(X > 39.5)$ instead. I am not an expert on this topic so I won't comment on whether or not this is reasonable or not. I just wanted to point out that this could be an explanation.

As a final remark, note that $P(X > 39.5) \approx P(X > 40.5)$ as long as $np$ is not too close to 40. Hence in a some situations it does not matter which approach you use (as you are using approximations anyway).

Sources:

Continuity correction: https://en.wikipedia.org/wiki/Continuity_correction

Edit: It just occured to me that your professor might intepret it as greater or equal to 40 in which case the probability of interest is $P(X \geq 40)=P(X>39)$. Then you would both agree and pick $39.5$.