A recent survey by the New Statesman on British social attitudes asked respondents if they believe that inequality is too large. The survey found that 74% of the respondents do believe inequality is too large. In a sample of six British citizens, what is the probability that two believe inequality is too large? In a sample of six British citizens, what is the probability that at least two respondents believe that inequality is too large? In a sample of four British citizens, what is the probability that none believe inequality is too large?

(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:

(a)

Compute the probability that in a a sample of six British citizens two believe inequality is too large as follows:

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)

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:

Thus, the probability that in a a sample of four British citizens none believe inequality is too large is 0.0046.