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 4 randomly selected consumers. Complete parts (a) through (d) below.

a. What is the probability that exactly 4 of the selected consumers recognize the brand name?

The probability that exactly 4 of the 5 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 4 of the selected consumers recognize the brand name?

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

(Round to three decimal places as needed.)

d. If 5 consumers are randomly selected, is 4 an unusually high number of consumers that recognize the brand name?

Since the value of probability is fixed (50%), there are only 2 possible outcomes (Recognize or Not recognize), and number of trials is fixed (5 Trials) which are independent of each other, this is a problem of Binomial Distribution. All conditions which are required for an experiment to be considered Binomial are satisfied.

Part a) Probability of exactly 4 out of 5

The formula to calculate exactly x of the n successes in Binomial experiment is:

[tex]P(X=x) = ^nC_{x} (p)^{x}(q)^{n-x}[/tex]

Here, q = 1 - p = 1 - 0.50 = 0.50

For exactly 4 out of 5 success, n will be 5, x will be 4. Using these values, we get:

[tex]P(X=4)=^5C_{4}(0.5)^{4}(0.5)^{1}=0.15625[/tex]

The probability that exactly 4 of the 5 consumers recognize the brand name is 0.156

Part b) Probability of all 5

For this case, x will be 5, as all selected consumers recognize the brand name. Using the values again, we get:

[tex]P(X=5)=^5C_{5}(0.5)^{5}(0.5)^{0}=0.03125[/tex]

The probability that all of the selected consumers recognize the brand name is 0.031

Part c) Probability of atleast 4

Atleast 4 means 4 or greater than 4. This means the number of consumers who recognize the brand should be either 4 or 5.

[tex]P(X\geq 4)=P(X=4) + P(X=5)[/tex]

Using the values from the last two parts, we get:

[tex]P(X\geq 4)=0.156+0.031=0.187[/tex]

Thus, The probability that at least 4 of the selected consumers recognize the brand name is 0.187

Part d)

Since, the probability of atleast 4 consumers recognizing the brand names is not unusual (less than 0.05), 4 is not an unusually high number of consumers that recognize the brand name.