A brand name has a 60 recognition rate Assume the owner of t

A brand name has a 60% recognition rate. Assume the owner of the brand wants to verify that rate by beginning with a small sample of 4 randomly selected consumers. Complete parts (a) through (d) below.

a. What is the probability that exactly 3 of the selected consumers recognize the brand name?

The probability that exactly 3 of the 4 consumers recognize the brand name is ____.

(Round to three decimal places as needed.)

b. What is the probability that all of the selected consumers recognize the brand name?

The probability that all of the selected consumers recognize the brand name is ____.

(Round to three decimal places as needed.)

c. What is the probability that at least 3 of the selected consumers recognize the brand name?

The probability that at least 3 of the selected consumers recognize the brand name is ____.

(Round to three decimal places as needed.)

d. If 4 consumers are randomly selected, is 3 an unusually high number of consumers that recognize the brand name?

A. No, because the probability that 3 or more of the selected consumers recognize the brand name is greater than 0.05.

B. Yes, because the probability that 3 or more of the selected consumers recognize the brand name is less than 0.05.

C. No, because the probability that 3 or more of the selected consumers recognize the brand name is less

than 0.05.

D. Yes, because the probability that 3 or more of the selected consumers recognize the brand name is greater than 0.05.

Solution

Binomial Distribution

PMF of B.D is = f ( k ) = ( n k ) p^k * ( 1- p) ^ n-k
Where   
k = number of successes in trials
n = is the number of independent trials
p = probability of success on each trial

a)
P( X = 3 ) = ( 4 3 ) * ( 0.6^3) * ( 1 - 0.6 )^1
= 0.3456
b)
P( X = 4 ) = ( 4 4 ) * ( 0.6^4) * ( 1 - 0.6 )^0
= 0.1296
c)

P( X < 3) = P(X=2) + P(X=1) + P(X=0)
= ( 4 2 ) * 0.6^2 * ( 1- 0.6 ) ^2 + ( 4 1 ) * 0.6^1 * ( 1- 0.6 ) ^3 + ( 4 0 ) * 0.6^0 * ( 1- 0.6 ) ^4
= 0.5248
P( X > = 3 ) = 1 - P( X < 3) = 0.4752

d)
D. Yes, because the probability that 3 or more of the selected consumers recognize the brand name is greater than 0.05.