Respuesta :
The probability that the number of residents that recognize the brand is not 4 is 0.86 , the probability that at least 7 people will recognize the brand is 0.29 and the probability that at most 5 people will recognize the coffee shop is 0.39.
Given Recognition rate of brand in town of Coffleton is 57% and sample size is 10.
We have to use binomial distribution:
P(X=x)=[tex]C_{n,x} p^{x} (1-p)^{n-x}[/tex]
[tex]C_{n,x} =C!/r!(C-r)![/tex]
A) The parameters are as under:
x is the number of successes.
n is the number of trials which is 10 in this question.
p is the probability.
In this problem :
The brand name of a certain chain of coffee shops has 57% recognition rate in the town of coffleton , hence p=0.53.
There is a sample of 10 residents, n=10.
Probability that the no. of people that recognize the brand name is not 4 is as under:
P(X≠4)=1-P(X=4)
P(X=4)=[tex]C_{10,4}(0.53)^{4} (0.43)^{6}[/tex]
=0.1395
P(X≠4)=1-0.1395
=0.8605
B) P(X>=7)=[tex]C_{10,7} (0.57)^{7} (0.43)^{3} +C_{10,8} (0.57)^{8} (0.43)^{2} +C_{10,9} (0.57)^{9} (0.43)^{1} +C_{10,10} (0.57)^{10} (0.43)^{0}[/tex]
=0.18012+0.081+0.0258+0.0036
=0.29
C)P(X<=5)=[tex]C_{10,1} (0.57)^{1} (0.43)^{9} +C_{10,2} (0.57)^{2} (0.43)^{8} +C_{10,3} (0.57)^{3} (0.43)^{7} +C_{10,4} (0.57)^{4} (0.43)^{6} +C_{10,5} (0.57)^{5} (0.43)^{5}[/tex]
=0.00285+0.0146205+0.0432+0.126+0.21168
=0.3983.
Hence the probability that the number of residents that recognize the brand is not 4 is 0.86 , the probability that at least 7 people will recognize the brand is 0.29 and the probability that at most 5 people will recognize the coffee shop is 0.39.
Learn more about binomial distribution at https://brainly.com/question/9325204
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