The brand name of a certain chain of coffee shops has a 53% recognition rate in

Question

The brand name of a certain chain of coffee shops has a 53% recognition rate in the town of Coffleton. An executive from the company wants to verify the recognition rate as the company is interested in opening a coffee shop in the town. He selects a random sample of 10 Coffleton residents. a. Identify n and p. For each of the following questions, before solving the problem, represent the question symbolically (i.e. P(X> 5). Write the R-code and use it to answer the questions, b. Find the probability that the number that recognize the brand name is equal to 4. c. Find the probability that the number that recognize the brand name is 5 or more. d. Find the probability that the number that recognize the brand name is at most 4.

Explanation / Answer

a.
n=10,p=0.53
b.
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

P( X = 4 ) = ( 10 4 ) * ( 0.53^4) * ( 1 - 0.53 )^6
= 0.1786
c.
P( X > = 5 ) = 1 - P( X < 5) = 0.6943
d.
P( X < = 4) = P(X=4) + P(X=3) + P(X=2) + P(X=1) + P(X=0)   
= ( 10 4 ) * 0.53^4 * ( 1- 0.53 ) ^6 + ( 10 3 ) * 0.53^3 * ( 1- 0.53 ) ^7 + ( 10 2 ) * 0.53^2 * ( 1- 0.53 ) ^8 + ( 10 1 ) * 0.53^1 * ( 1- 0.53 ) ^9 + ( 10 0 ) * 0.53^0 * ( 1- 0.53 ) ^10
= 0.3057

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