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:Step-by-step 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 = [tex]E(x) = np = 240[/tex]Var(x) = [tex]npq = 182.4[/tex]Standard deviation = [tex]\sqrt{182.4} =13.51[/tex]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 CanadiansWith continuity correction this equals[tex]P(209.5<X<270.5)\\= P(\frac{209.5-240}{13.51} <Z<\frac{270.5-240}{13.51})\\=P(-2.26<Z<2.26)\\=2(0.4881)\\= 0.9762[/tex]