Respuesta :
Answer:
(a) The probability that in a a sample of six British citizens two believe inequality is too large is 0.0375.
(b) The probability that in a a sample of six British citizens at least two believe inequality is too large is 0.9944.
(c) The probability that in a a sample of four British citizens none believe inequality is too large is 0.0046.
Step-by-step explanation:
The random variable X can be defined as the number of British citizens who believe that inequality is too large.
The proportion of respondents who believe that inequality is too large is, p = 0.74.
Thus, the random variable X follows a Binomial distribution with parameters n and p = 0.74.
The probability mass function of X is:
[tex]P(X=x)={n\choose x}\ 0.74^{x}(1-0.74)^{n-x};\ x=0,1,2,3...n[/tex]
(a)
Compute the probability that in a a sample of six British citizens two believe inequality is too large as follows:
[tex]P(X=2)={6\choose 2}\ 0.74^{2}(1-0.74)^{6-2}\\=15\times 0.5476\times 0.00456976\\=0.03753600864\\\approx 0.0375[/tex]
Thus, the probability that in a a sample of six British citizens two believe inequality is too large is 0.0375.
(b)
Compute the probability that in a a sample of six British citizens at least two believe inequality is too large as follows:
P (X ≥ 2) = 1 - P (X < 2)
= 1 - P (X = 0) - P (X = 1)
[tex]=1-[{6\choose 0}\ 0.74^{0}(1-0.74)^{6-0}]-[{6\choose 1}\ 0.74^{1}(1-0.74)^{6-1}]\\\\=1-[1\times 1\times 0.000308915776]-[6\times 0.74\times 0.0011881376]\\\\=1-0.00031-0.0053\\\\=0.99439\\\\\approx 0.9944[/tex]
Thus, the probability that in a a sample of six British citizens at least two believe inequality is too large is 0.9944.
(c)
Compute the probability that in a a sample of four British citizens none believe inequality is too large as follows:
[tex]P(X=0)={4\choose 0}\ 0.74^{0}(1-0.74)^{4-0}\\=1\times 1\times 0.00456976\\=0.00456976\\\approx 0.0046[/tex]
Thus, the probability that in a a sample of four British citizens none believe inequality is too large is 0.0046.
