Question

Despite a slew of controversies and 18 months of constant rioting, President Macron of France has managed to bring his approval rating up to a staggering 27% (Up 1% from when I asked this question last semester). Suppose you were to independently and randomly sample 159 French citizens with replacement and ask them whether or not they approve of President Macron’s performance.

(a) Write down the equation, in terms of x, for the probability a certain number of people x within your sample approve of president Macron.

(b) Is it feasible to use this equation in (a) to compute probabilities?

(c) Can you use an alternative method to approximate the probability of a certain number of people in your sample approve of president Macron? State the conditions which you must meet in order to do so, and whether or not they are met.

(d) If you meet the conditions for approximation in (c), use the approximation to calculate the probability of between 25 and 60 (inclusive) people in your sample approve of president Macron. Be sure to apply the appropriate corrections to your approximation, if needed.

Answer 1

Let x be the number of people approve of president Macron.

x follows binomial distribution with n = 159 and p = 0.27

a) P( x ) = _nC_x * p^x * q^(n-x) ; x = 0 ,1,2,3...n and q = 1- p = 1-0.27 = 0.73

b) It is not feasible to use this equation to compute probabilities because n has large value.

c) Yes, we can use normal approximation to compute the probability of a certain number of people in your sample approve of president Macron.

if n*p and n*q both are greater than 5 , then we use the normal approximation

Therefore n*p = 159*0.27 = 42.93 and n*q = 159*0.73 = 116.07  

Conditions are met ,both n*p and n*q both are greater than 5 , we can use the normal approximation .

d) We are asked to find P( 25 ≤ x ≤ 60) using normal approximation.

Since we are using normal approximation , we need to use continuity correction by subtracting 0.5 from 25 and adding 0.5 to 60

P( 24.5 ≤ x ≤ 60.5 )

Now we have to find mean ( µ ) = n*p = 42.93

and standard deviation ( σ ) = = = 5.5981

P( 24.5 ≤ x ≤ 60.5 ) = P( x ≤ 60.5 ) - P( x ≤ 24.5 )

=

= P( z < 3.14 ) - P( z < -3.29 )

= 0.9992 - 0.0005 ----- ( from z score table )

= 0.9987

Probability of between 25 and 60 (inclusive) people in your sample approve of president Macron is 0.9987