At a high school with 800 students, 80% of the students ride the school bus. if 20 students are selected randomly (without replacement) and we let x = the number of students in the sample who ride the bus, then x does not exactly have a binomial distribution. why is it nevertheless appropriate to approximate probabilities for x using the binomial distribution for n = 20 and p = 0.8? the binomial is always appropriate when sampling without replacement. because the sample is less than 10% of the population, it is appropriate to use the binomial distribution even though the samples are not strictly independent. since np > 10, we can still use the binomial distribution. next
Since sample is less than 10% of the population, we can use binomial distribution to approximate the probability.
So we want to calculate P( X < 20 ).
We can calculate the probability of complement event P( X = 20 ).
Use binomial formula b(x; n, p) = C( n, x ) p^(x) (1-p)^(n-x)
b( 20; 20, 0.8 )
= C( 20, 20 ) (0.8)^(20) (0.2)^0
= (0.8)^20 ≈ 0.0115
So since P( X < 20 ) = 1 - P( X = 20 ), we have
P( X < 20 ) ≈ 1 - 0.0115 = 0.9885