Q1. a) Assume that the distribution of birthday during the months of the year is uniform. What is the probability that a class of 120 students, exactly 20 students have their birthdays in either August or September ? Solving using
i. the exact Binomial distribution
ii. the Normal approximation to the Binomial distribution.
Not sure how to write out my answer but I basically used the formula for the Binomial distribution $\binom nkp^k (1-p)^{n-k}$ Using $p= \frac{120}{365}, \ n=120, \ k=20$.
I have been sitting on a few stats problem since a couple of days and I have tried various ways so if you can help I will appreciate it !
Your binomial formula is correct but $p$ should be $\frac16$.
Your value for $p$ is incorrect, firstly because August and September combine for $62$ days, not $120$, and secondly because the question asks you to treat the months as being distributed uniformly rather than according to their number of days. Since August and September are $2$ of the $12$ months, $p=\frac16$.
The formula for the using the normal approximation is similar to the typical $z$-formula for proportions, and uses the proportion and standard error of the binomial distribution. Since we are using a continuous distribution to approximate a discrete one, we would use an integer correction and look for the probability that $19.5<X<20.5$ You would then find the $p$-value for both $19.5$ and $20.5$ using the $z$-test with the following formula:$$Z={\bar X-\mu\over \sigma}$$ where $\mu=np$ and $\sigma = \sqrt{np(1-p)}$