Question

According to the 2010 census, 4.0% of U.S. households were multigenerational (containing at least three generations: grandparents, parents, and children). A more recent study found that among 580 U.S. households surveyed, 28 were multigenerational. Does this study provide strong enough evidence to conclude that the percentage of multigenerational households has changed

Answer 1

We know that at CI of 90% the new study provide evidence to claim that the porcentage of multigenerational households has not change

Explanation:

To use Normal Distribution as an approximation of the binomial distribution

and develop a test of proportion we look at the products:

n*p and n*q in the sample

p = 28/580 p = 0,04827 then q = 0,95173

and n*p = 580*0,04827 = 27,99 > 5

n*q = 580*0,95173 = 552 > 5

Hypothesis Test:

p₀ = 4 % p₀ = 0,04

sample size n = 580

p sample proportion mean p = 0,04

Chossing a confidence Interval CI = 90 % then α = 10% α = 0,1

α/2 = 0,05 and z(score) = 1,64 z (c) = 1,64

The test is a doble tail test since the question is about difference between the study and the past census

Null Hypothesis H₀ p = p₀

Alternative Hypothesis Hₐ p ≠ p₀

To calculate z(statistics) z(s)

z(s) = ( p - p₀ ) / √ p*q/n

z(s) = ( 0,04827 - 0,04 ) * √580 / √0,04827*0,95173

z(s) = 0,00827*24,08/ √0,04594

z(s) = 0,1991/ 0,2143

z(s) = 0,929

Comparing z(s) and z(c)

z(s) < z(c) ; 0,929 < 1,64

Therefore at CI 90% we accept H₀ since z(s) is in the acceptance region