6) A survey on British Social Attitudes asked respondents if they had ever boyco

Question

6) A survey on British Social Attitudes asked respondents if they had ever boycotted goods for ethical reasons (Statesman, January 28, 2008). The survey found that 34% of the respondents have boycotted goods for ethical reasons.

a) In a sample of six British citizens, what is the probability that two have ever boycotted goods for ethical reasons?

b) In a sample of six British citizens, what is the probability that at least two respondents have boycotted goods for ethical reasons?

c) In a sample of ten British citizens, what is the probability that none have boycotted goods for ethical reasons?

d) In a sample of ten British citizens, what is the expected number of people that have boycotted goods for ethical reasons?

Explanation / Answer

a)

Note that the probability of x successes out of n trials is          
          
P(n, x) = nCx p^x (1 - p)^(n - x)          
          
where          
          
n = number of trials =    6      
p = the probability of a success =    0.34      
x = the number of successes =    2      
          
Thus, the probability is          
          
P (    2   ) =    0.329021922 [ANSWER]

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b)

Note that P(at least x) = 1 - P(at most x - 1).          
          
Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    6      
p = the probability of a success =    0.34      
x = our critical value of successes =    2      
          
Then the cumulative probability of P(at most x - 1) from a table/technology is          
          
P(at most   1   ) =    0.338129796
          
Thus, the probability of at least   2   successes is  
          
P(at least   2   ) =    0.661870204 [ANSWER]

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c)

Note that the probability of x successes out of n trials is          
          
P(n, x) = nCx p^x (1 - p)^(n - x)          
          
where          
          
n = number of trials =    10      
p = the probability of a success =    0.34      
x = the number of successes =    0      
          
Thus, the probability is          
          
P (    0   ) =    0.015683369 [ANSWER]

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d)

E(x) = n p = 10*0.34 = 3.4 [ANSWER]

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