Question

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 facts about the population. About 24% of the Canadian population over 15 years of age are first generation; that is, they were born outside Canada. The number X of first-generation Canadians in random samples of 1000 persons over 15 should therefore vary with the binomial (n = 1000, p = 0.24) distribution.

(a) What are the mean and standard deviation of X?

(b) Use the Normal approximation to find the probability that the sample will contain between 210 and 270 first-generation Canadians. Be sure to check that you can safely use the approximation.

Answer 1

Explanation:

Given that about 24% of the Canadian population over 15 years of age are first generation; that is, they were born outside Canada.

X - first-generation Canadians in random samples of 1000 persons over 15 should therefore vary with the binomial (n = 1000, p = 0.24) distribution.

a) Mean of X =
`E(x) = np = 240`

Var(x) =
`npq = 182.4`

Standard deviation =
`√(182.4) =13.51`

b) When approximated to normal this variable X will be normal

we check whether np and nq are greater than 5.

Here we find both are greater than 5. So binomial to normal approximation can be done.

X is N(240, 13.51)

the probability that the sample will contain between 210 and 270 first-generation Canadians

With continuity correction this equals

`P(209.5<X<270.5)\\= P((209.5-240)/(13.51) <Z<(270.5-240)/(13.51))\\=P(-2.26<Z<2.26)\\=2(0.4881)\\= 0.9762`