Binomial Distribution Excel Function Probability Problem

Question

Here's the question: According to the CDC, the proportion of American women, aged 18-24, that binge drink is $p=0.24$. Your college surveys a SRS of 200 female students and finds that 56 binge drink.

If the proportion of women on your campus who binge drink is the same as the national proportion, what is the probability that the number of binge drinkers in a SRS of $200$ students is as equal or larger than the result of the college's sample? (Round to 3 decimal places)

I got $.918$ as my answer but it's wrong.

How do I get the correct answer? I'm using the =binom.idst() function in Excel to solve the problem.

Answer 1

In excel BINOMDIST$(k;200;0.24;FALSE)$ tells the probability that in $200$ experiment $k$ are successful given that the probability of a success is $0.24$.

I've got:

$$\sum_{k=56}^{200} \text{ BINOMDIST}(k;200;0.24;FALSE)=0.10843$$

for the probability that in $200$ cases $56$ or more are binge drinkers.

Also, BINOMDIST$(k;200;0.24;TRUE)=P(\text{number of successes }\le k).$

So,

$$P(\text{number of successes }\ge 56)=1-\text{BINOMDIST}(55;200;0.24;TRUE)=0.10843.$$

Answer 2

First, assuming that the true proportion of female binge drinkers at your college is $p = 0.24$, then the number $X$ of such binge drinkers in a sample of size $n = 200$ would be approximately binomially distributed: $$X \sim \operatorname{Binomial}(n = 200, p = 0.24),$$ with $$\Pr[X = x] = \binom{n}{x} p^x (1-p)^{n-x} = \binom{200}{x} (0.24)^x (0.76)^{200-x}.$$ So the probability that you would observe a sample in which the number of female binge drinkers is at least as large as the sample you obtained is the probability that $X \ge 56$: $$\Pr[X \ge 56] = \sum_{x=56}^{200} \binom{200}{x} (0.24)^x (0.76)^{200-x}.$$ You could calculate this on a computer, but in Excel, you would use =1-BINOM.DIST(55,200,0.24,TRUE) because Excel calculates only the cumulative distribution function; i.e., $\Pr[X \le x]$, rather than $\Pr[X \ge x]$. So you have to calculate a complementary probability: $$\Pr[X \ge 56] = 1 - \Pr[X \le 55].$$ The TRUE argument at the end tells Excel to calculate the CDF rather than the PDF.