2 methods of gives me different numbers of questions to be guessed correct using the same p value.

Question

I have a test with 30 multiple choice questions each with 3 possible answers. I calculated the number of questions that need to be guessed correctly at a probability level of p=0.0455 (2 standard deviations). I did this 2 ways:

First, I added 2 standard deviations to the mean. The mean is np=10. The STD is sqrt(np(1-p))=2.5820. So, 10+2(2.5820)=15.164. (I know I can't answer 15.164 questions correctly, but I'm only interested in comparing the outcome of this method to my next method.)

Second, I added all the possibilities of getting 14 or fewer questions correct and subtracted from one. That should give me the probability of getting 15 or more correct. So, p(k)=(30 choose k)(p^k)(1-p)^(n-k) where k is the number of correct answers. I then summed p(k) from k=0 to k=14 and subtracted from one. This gives me 0.043. That means that I need slightly fewer than 15 correct answers to get a p=0.0455.

So, one way I get 15.2 correct answers and the other way I get slightly fewer than 15 correct answers for the same p value of 0.0455.

My computer has at least 7 decimals of accuracy, so why are these answers so different?

Note: I set up my computer spreadsheet to do each of these calculations for a project and noticed the difference. I wondered if I was doing something wrong.

Answer 1

There are a variety of issues here in your attempt to use a normal approximation to a binomial distribution

1. You should not have used $2$ (or even $1.96$) to get an $95\%$ probability on a normal distribution. That is for two-sided calculations, and is in fact the point $97.5\%$ up the distribution so there is a $95\%$ probability to be $\pm1.96$ standard deviations from the mean. But one-sided you should use something like $1.644854$. Your normal approximation would have given about $14.25$ if calculated this way.

2. The normal distribution is continuous while the binomial distribution is discrete. If you draw the cumulative distribution functions and look at the relevant part, they look like this

So the probability of $14$ or fewer correct is (using your calculations or the red binomial line) about $0.9565$, and so of $15$ or more correct about $0.0435$.

3. Since you can only get an integer number of correct answers, it is worth looking at the probability of $14.5$ or less with your normal approximation (the $.5$ being called a continuity correction), about $\frac{14.5-10}{2.582}\approx 1.74$ standard deviations above the mean. If you did this (using tables or the black normal line) it would suggest about $0.9593$, and so a probability of about $0.0407$ for more than $14.5$, not exact but not bad for an approximation.

Either way, the conclusion would be that you have a probability of between $4\%$ and $5\%$ of getting $15$ or more questions correct.