One way of checking the effect of undercoverage nonresponse

One way of checking the effect of undercoverage, nonresponse, and other sources of error in a sample survey is to compare the sample with known demographic facts about the population. The 2000 census found that 23,772,494 of the 209,128,094 adults (aged 18 and over) in the United States called themselves \"Black or African American.\"

(a) What is the population proportion p of blacks among American adults? (Round your answer to four decimal places.)


(b) An opinion poll chooses 1236 adults at random. What is the mean number of blacks in such samples? (Round your answer to one decimal place.)
blacks

(c) Use a Normal approximation to find the probability that such a sample will contain 122 or fewer blacks. Be sure to check that you can safely use the approximation. (Round your answer to four decimal places.)

Solution

(a) population proportion p =23772494/209128094 = 0.1137

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(b) mean=n*p=1236*0.1137 =140.5

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(c) standard deviation =sqrt(n*p*(1-p))

=sqrt(1236*0.1137*(1-0.1137)) =11.1604

So the probability is

P(X<122) = P((X-mean)/s <(122-140.5)/11.1604)

=P(Z<-1.66) =0.0485 (from standard normal table)