Question

A brand name has a 50% recognition rate. Assume the owner of the brand wants to verify that rate by beginning with a small sample of 8 randomly selectedconsumers. Complete parts (a) through (d) below:

a. What is the probability that exactly 7 of the selected consumers recognize the brand name?

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

c. What is the probability that at least 7 of the selected consumers recognize the brand name? (round to three decimal places)

d. If 8 consumers are randomly​ selected, is 7 an unusually high number of consumers that recognize the brand​ name?

A. Yes​, because the probability that 7 or more of the selected consumers recognize the brand name is greater than 0.05.

B.No​, because the probability that 7 or more of the selected consumers recognize the brand name is greater than 0.05.

C. No because the probability that 7 or more of the selected consumers recognize the brand name is less than 0.05.

D.Yes​, because the probability that 7 or more of the selected consumers recognize the brand name is less than 0.05.

Answer 1

a. The probability that exactly 4 of the 5 consumers recognize the brand name is

P(X=7) = (⁸C₇)* (0.50)⁷ * (0.50)^8-7 = 8 * 0.5⁷ * 0.5¹ = 0.03125 ~ 0.0313

the probability that exactly 7 of the selected consumers recognize the brand name is 0.031.

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

P(X =8) = (0.50)⁸

= 0.0039

The probability that all of the selected consumers recognize the brand name is 0.0039.

c. The probability that at least 4 of the selected consumers recognize the brand name

P( X >=7) = P(X= 7 or X=8)

= 0.0313 + 0.0039

= 0.0352

The probability that at least 4 of the selected consumers recognize the brand name is 0.0352.

d.

Yes​, because the probability that 7 or more of the selected consumers recognize the brand name is less than 0.05.

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