Question

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 nothing. ​(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 nothing. ​(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 nothing. ​(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. Yes​, 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. No​, because the probability that 3 or more of the selected consumers recognize the brand name is greater than 0.05.

Answer 1

P( regonition rate) = 0.6 = p

n= 4

Let X be the number of people who recognize the brand

X~ Binomial ( 4, 0.6)

a) probability that exactly 3 of the selected consumers recognize the brand​ name

= P( X=3)

= 4C3 ( 0.6)3 (1-0.6)

= 0.3456

b)  probability that all of the selected consumers recognize the brand​ name

= P( X=4)  

= (0.6)4

= 0.1296

c)  probability that at least 3 of the selected consumers recognize the brand​ name

P( X>=3)

= P( X=3) + P(X=4)

=0.3456 + 0.1296

=0.4752

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