Question

Some of the methods in this section are approximations rather than exact probability results. We have given rules of thumb for safe use of these approximations. (a) You are interested in attitudes toward drinking among the 75 members of a fraternity. You choose 30 members at random to interview. One question is "Have you had five or more drinks at one time during the last week?" Suppose that in fact 30% of the 75 members would say "Yes." Explain why you cannot safely use the B(30, 0.3) distribution for the count X in your sample who say "Yes." The population (the 75 members of the fraternity) is only 2.5 times the size of the sample. Our rule of thumb says that this ratio should be at least _______________.

(b) The National AIDS Behavioral Surveys found that 0.2% (that's 0.002 as a decimal fraction) of adult heterosexuals had both received a blood transfusion and had a sexual partner from a group at high risk of AIDS. Suppose that this national proportion holds for your region. Explain why you cannot safely use the Normal approximation for the sample proportion who fall in this group when you interview an SRS of 1000 adults. Our rule of thumb for the Normal approximation calls for np and n(1 − p) to be at least __________; we have np = (1000)(0.002) = 2.

Answer 1

(A) In this case, sample size is 30 which is more 10% of population size because

(30/75)*100 = 40%

So, we know that the maximum sample size allowed is equal to 10% of the population size, but not more than 10%

Therefore, Our rule of thumb says that this ratio should be at least 10 times of the size of the sample

(B) Rule of thumb for normal approximation is that np and n(1-p), both must be at least 10 or more

In this case, np = 1000*0.002 = 2. So, this is not valid to assume normal approximation...