Answer:
a
[tex]P(X = 2) = 0.2547[/tex]
b
[tex]P(X > 2) = 0.5623 [/tex]
c
[tex]P(2 \le X \le 5 ) = 0.7817 [/tex]
Step-by-step explanation:
From the question we are told that
The proportion of US adults who say that they are likely to make purchase during a sale tax holiday is
[tex]p = 0.28[/tex]
The sample size is n = 10
Generally the distribution of US adults who say that they are likely to make purchase during a sale tax holiday, follows a binomial distribution
i.e
[tex]X \~ \ \ \ B(n , p)[/tex]
and the probability distribution function for binomial distribution is
[tex]P(X = x) = ^{n}C_x * p^x * (1- p)^{n-x}[/tex]
Here C stands for combination hence we are going to be making use of the combination function in our calculators
Generally the probability number of adult is exactly 2 is mathematically represented as
[tex]P(X = 2) = ^{10}C_2 * (0.28)^2 * (1- 0.28)^{10-2}[/tex]
=> [tex]P(X = 2) = 45 * 0.0784 * 0.0722[/tex]
=> [tex]P(X = 2) = 0.2547[/tex]
Generally the probability number of adult is more than 2 is mathematically represented as
[tex]P(X > 2) = 1- [P(X \le 2) ][/tex]
[tex]P(X > 2) = 1- [[P(X = 0)] +[P(X = 1) ] + [P(X = 2)] ][/tex]
=> [tex]P(X > 2) = 1- [^{10}C_0 * (0.28)^0 * (1- 0.28)^{10-0}] +[^{10}C_1 * (0.28)^1 * (1- 0.28)^{10-1}] + [^{10}C_2 * 0.28^2 * (1- 0.28)^{10-2}] ][/tex]
=> [tex]P(X > 2) = 1- [0.0374 + 0.1456 + 0.2547 [/tex]
=> [tex]P(X > 2) = 0.5623 [/tex]
Generally the probability number of adult is between two and five, inclusive is mathematically represented as
[tex]P(2 \le X \le 5 ) = [P(X = 2 ) + P(X = 3 ) +P(X = 4) + P(X= 5 )] [/tex]
=> [tex]P(2 \le X \le 5 ) = [[^{10}C_32 * 0.28^2 * (1- 0.28)^{10-2}] + [^{10}C_3* 0.28^3 * (1- 0.28)^{10-3}] +[^{10}C_4 * 0.28^4 * (1- 0.82)^{10-4}] + [^{10}C_5 * 0.28^5 * (1- 0.28)^{10-5}]] [/tex]
=> [tex]P(2 \le X \le 5 ) = [[0.2547] + [120* 0.02195 * 0.1003] +[210 * 0.00615 * 0.1393] + [252 * 0.0017 * 0.1935]] [/tex]
=> [tex]P(2 \le X \le 5 ) = [[0.2547] + [0.2642] +[0.1799] + [0.0829] [/tex]
=> [tex]P(2 \le X \le 5 ) = 0.7817 [/tex]
