A major coca company sponsors a national taste test, in which subjects sample its cola as well as the best-selling brand. Neither is identified by brand, they're asked to choose which one tastes better. We survey 25 people, who have no preference, and arbitrarily choose one of the colas. Find the probability that 15 or more choose the cola from the sponsor company.
My attempt:
mean = n*p= (25)(.5)=12.5
standard deviation = $\sqrt{n*p*q} = \sqrt{25*.5*.5} = \sqrt {6.25} = 2.5$
z= (15-12.5)/2.5= 1
Prob(x>15) = $1-\int^1_{-\infty} \frac{1}{\sqrt{2\pi}}e^{-.5x^2} dx = .1586 $
But these are none of the answer choices available, so I guess I did something wrong. Any help is appreciated!
Answers choices that are given are: .0983, .2830, .1847, .3731, .2119
$X$ is a discrete variable. Therefore $P(X\geq 15)=1-P(X\leq 14)$
You want to apply the central limit theorem. With the continuity correction factor ($+0.5$) the equation is
$$P(X\geq 15)\approx 1-\Phi\left( \frac{14+0.5-12.5}{2.5} \right)=1-\Phi(0.8)$$
Note that this is an approximation.
Using integral representation it becomes
$$P(X\geq 15)\approx 1-\int^{0.8}_{-\infty} \frac{1}{\sqrt{2\pi}}e^{-0.5\cdot x^2} dx$$