Suppose that a random sample X1,...,Xn is to be taken from the uniform distribution on the interval [0, θ ] and that θ is unknown. How large a random sample must be taken in order that

Difference is (0.181671, 0.263529)

We are more than 95% confident that the true percentage of life births for women under 38 years of age is between 18.17% and 26.35% higher than the true population of live births of women aged 38 or older.

Step-by-step explanation:

We are given;

x1 = 41

n1 = 152

x2 = 4

n2 = 86

Thus,

Sample proportion 1; p'1 = 41/152 = 0.2697

Sample proportion 2; p'2 = 4/85 = 0.0471

For confidence level, 1 - α = 1 - 0.05 = 0.95, using the z-table i attached under the column of z_α/2,we have

z_α/2 = 0.96

The end points of the confidence intervals for p1 - p2 are;

(p'1 - p'2) - z_α/2√[(p'1(1 - p'1)/n1) + (p'2(1 - p'2)/n2)

And

(p'1 - p'2) + z_α/2√[(p'1(1 - p'1)/n1) + (p'2(1 - p'2)/n2)

The first one is calculated as;

(0.2697 - 0.0471) - 0.96√[(0.2697(1 - 0.2697)/152) + (0.0471(1 - 0.0471)/86)

.= 0.2226 - 0.040929 = 0.181671

The second one is calculated as;

(0.2697 - 0.0471) + 0.96√[(0.2697(1 - 0.2697)/152) + (0.0471(1 - 0.0471)/86)

.= 0.2226 + 0.040929 = 0.263529

Thus, we are more than 95% confident that the true percentage of life births for women under 38 years of age is between 18.17% and 26.35% higher than the true population of live births of women aged 38 or older.