An article regarding interracial dating and marriage recently appeared in a news

Question

An article regarding interracial dating and marriage recently appeared in a newspaper. Of the 1718 randomly selected adults, 306 identified themselves as Latinos, 321 identified themselves as blacks, 251 identified themselves as Asians, and 777 identified themselves as whites. Among Asians, 79% would welcome a white person into their families, 71% would welcome a Latino, and 66% would welcome a black person.

NOTE: If you are using a Student's t-distribution, you may assume that the underlying population is normally distributed. Construct the 95% confidence intervals for the three Asian responses.

Explanation / Answer

Back-up Theory

100(1 - )% Confidence Interval (CI) for population proportion, p, is:

pcap ± Z/2SE(pcap)

= pcap ± Z/2{pcap(1 - pcap)/n}, where pcap = sample estimate of p, Z/2 = upper (/2) percent point of N(0, 1) and n = sample size.

Now, to work out the solutions,

Let p1, p2 and p3 be respectively = population proportion of Asians who would welcome into their families a white person, a Latino, and a black person.

Then, from the given data, n = 251, p1cap = 0.79, p2cap = 0.71, p3cap = 0.66.

Employing the formula given under Back-up Theory, 95% Confidence Interval (CI) are:

CI for p1 = 0.79 ± 1.96{0.79(0.21)/251} = 0.79 ± 0.0504 = [0.7396, 0.8404] ANSWER 1

CI for p2 = 0.71 ± 1.96{0.71(0.29)/251} = 0.71 ± 0.0561 = [0.6539, 0.7661] ANSWER 2

CI for p3 = 0.66 ± 1.96{0.66(0.34)/251} = 0.66 ± 0.0586 = [0.6014, 0.7186] ANSWER 3

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