fifty percent of all americans have sedentary lifestyle by n

fifty percent of all americans have sedentary lifestyle by not exercising regularly. A random sample of 25 Americans are selected:

using central limit theorem approximate the probability that between 10 and 15 do not exercise regularly.

Solution

Normal Approximation to Binomial Distribution
Mean ( np ) =25 * 0.5 = 12.5
Standard Deviation ( npq )= 25*0.5*0.5 = 2.5
Normal Distribution = Z= X- u / sd                   
To find P(a < = Z < = b) = F(b) - F(a)
P(X < 10) = (10-12.5)/2.5
= -2.5/2.5 = -1
= P ( Z <-1) From Standard Normal Table
= 0.15866
P(X < 15) = (15-12.5)/2.5
= 2.5/2.5 = 1
= P ( Z <1) From Standard Normal Table
= 0.84134
P(10 < X < 15) = 0.84134-0.15866 = 0.6827