Question

Gluten sensitivity affects approximately 15% of people in the us.You decide to randomly sample 800 individuals and have them report their gluten intolerance. 96 people report gluten sensitivity. 

Answer 1

Let we have to calculate the probability that 115 individual in the sample with wheat intolerance.

Probability of wheat intolerance is p = 0.15

Sample size, n = 800

As np = 800$\times$ 0.15 = 120 > 10 and n(1-p) = 800$\times$ (1-0.15) = 680 > 10,

we can use the normal approximation to the binomial distribution with continuity correction.

a. Let X be the number of individuals in the sample with wheat intolerance.

X~N(np, np(1-p) or X~N(120, 102)

P(X = 115) = P(114.5 < x<115.5)

$=P(\dfrac{(114.5-120)}{\sqrt{102}}< z< \dfrac{(115.5-120)}{\sqrt{102}})$

$=P(-0.5446< z< -0.4456)$

= 0.293