Answer to Question #222341 in Statistics and Probability for kimutai

Question

Mathdrink a softdrink company,knows that it has 42% market share in one region of the province.MathDrink's marketing department conducts a blind test of 70 people at the local mall

i)Why is it safe to assume the normal distribution to binomial here?

ii)What is the probability that fewer than 25 people will choose MathDrink?

iii)What is the probability that exactly 25 people will choose MathDrink?

Answer 1

i and ii)

We have n=70, with p=0.42 and q=0.58.

Since n*p=70*0.42=29.4>5 and n*q=70*0.58=40.6>5 then we can use the normal

approximation.

If n*p was less than 5 or equaled 5, we could use the formula of binominal distribution to reduce time to calculate and increase the accuracy

mean=29.4 and S.D.=4.13

So, P(X<25) can be approximated to P(X<=24.5).

P(X<24.5)=P(Z<(24.5 - 29.4)/4.13)=P(Z<-1.19)=0.117

So, there is approx. 12% probability that fewer than 25 people surveyed will choose MathDrink.

iii)

P(X=25) can be approximated to P(24.5<X<25.5).

P(24.5<X<25.5)=P(-1.19<Z<-0.94)=P(Z<-0.94)-P(Z<-1.19)=0.057

The probability that exactly 25 people will choose MathDrink is approximately 5.7%.