in a random sample of 31 professional actors it was found th

in a random sample of 31 professional actors, it was found that 24 were extroverts, find a 98% confidence interval for p A. 0.647 to 1.485 B. 0.647 to 0.902 C. 0.589 to 0.959 D. 0.589 to 1.485 E. 0.063 to 0.959 Please specify which answer choice A, B, C, D, or E

Solution

You want to compute an interval so that you are 98% confident that true value p falls within this interval (p being the probability that someone is an extrovert).

Your sample estimate is 24/31 = .774 which I will call phat

You are drawing values from a binomial distribution which is the distribution of values where there is a probability of falling into one category or another, like flipping a coin that comes up heads or tails.

There are several ways of computing the 98% confidence interval (see http://en.wikipedia.org/wiki/Binomial_pr...

The easy way is to assume that the binomial distribution is approximately normal. Your sample of 31 should be close enough to make this approximation valid. By the central limit theorem the binomial distribution approaches the normal distribution as your sample size grows larger and larger.

The formula for a confidence interval using this approximation is:

phat +/- zcrit * sqrt ( phat* (1-phat) / n )

where n = sample size and zcrit is the critical value of the normal distribution that leaves the appropriate percentage of area in the tails. For a 98% confidence interval, you need to leave 2% (1-98%) of the area in the tails, so 1% in each tail.

You can find the value using this applet : http://davidmlane.com/hyperstat/z_table....
the zcrit is: 2.3263

sqrt ( (phat * (1-phat) / n) ) = .0671

So the confidence interval is phat +/- 2.3263*.0671 = (0.6, .9121)

To find the exact confidence interval (this is the hard way and is called the clopper pearson interval), you need to find the binomial coefficients such that the probability of observing less than 24 successes is .01 and greater than 24 successes is .01. It turns out that these probabilities are .595 and .892. Thus the actual interval is (.595, .892).