Statistics Confidence Interval

Question

Europeans have been more skeptical than Americans about the use of genetic engineering to improve foods. A sample survey gathered responses from random samples of 862 Americans and 12083 Europeans. (The European sample was larger because Europe is divided into many nations.) Subjects were asked to consider the following issue: 

Using modern biotechnology in the production of foods, for example to make them higher in protein, keep longer, or change in taste. 

They were asked if they considered this "risky for society." In all, 448 of Americans and 7628 of Europeans thought the application was risky.

Give a 97% confidence interval for the percent difference (±0.1) between Europe (call them group 1) and the United States:

Answer 1

Let n₁ = the number (sample size) of Europeans surveyed = 12083

Let n₂ = the number (sample size) of Americans surveyed = 862

Let x₁ = the number of Europeans who thought genetic engineering was risky = 7628

Let x₂ = the number of Americans who thought genetic engineering was risky = 448

Let p₁ = the proportion of Europeans who thought genetic engineering was risky

p₁ = (x₁)÷(n₁) = 7628 ÷ 12083 = 0.631

Let p₂ = the proportion of Americans who thought genetic engineering was risky

p₂ = (x₂)÷(n₂) = 448 ÷ 862 = 0.520

Even though this is considered a binomial random variable, the normal distribution may be used as a sampling distribution because n₁p₁ and n₂p₂ are both greater than 5.

The standard error (SE) is going to be [(p₁q₁ ÷ n₁)+((p₂q₂ ÷ n₂)]^1/2 = √(p₁q₁ ÷ n₁)+((p₂q₂ ÷ n₂)

Note that q₁ = (1-p₁) and q₂ = (1-p₂). Stated another way, q₁ is the proportion of Europeans who did NOT find genetic engineering risky and q₂ is the proportion of Americans who did NOT find genetic engineering risky.

The estimator for the difference in proportions of persons who found genetic engineering risky is going to be (p₁ - p₂).

For a 97% confidence interval, the critical Z value is 2.17.

The 97% CI for difference in proportions is

(p₁ - p₂) ± 2.17√(p₁q₁ ÷ n₁)+((p₂q₂ ÷ n₂)=(0.631-0.520)±√[ (0.631 x 0.369 ÷ 12083) + (0.520 x 0.480 ÷ 862)]

=0.111 ± 0.038

97% CI for difference in proportions (Europe - United States) is 0.073 to 0.149

97% CI for difference in PERCENTAGE is 7.3% to 14.9%