Independent samples of n1 = 400 and n2= 400 observations were selected from binomial populations 1 and 2, and x1 = 100 and x2 = 127 successes were observed. Calculate the approximate standard error for the statistic, the point estimator for the difference ( p1-p2) in the two binomial proportions?

A. 0.0441
B. 0.0287
C. 0.0536
D. 0.0318

To calculate the approximate standard error for the statistic and the point estimator for the difference (p1 - p2) in the two binomial proportions, we can use the following formula:

Standard Error = sqrt((p1 * (1 - p1) / n1) + (p2 * (1 - p2) / n2))

Where:
- p1 and p2 are the proportions of successes in populations 1 and 2 respectively,
- n1 and n2 are the sample sizes of populations 1 and 2 respectively.

Given:
- n1 = 400
- n2 = 400
- x1 = 100 (number of successes in sample 1)
- x2 = 127 (number of successes in sample 2)

To estimate p1 and p2, we'll use the following formulas:

p1 = x1 / n1
p2 = x2 / n2

Substituting the given values into the formulas, we get:

p1 = 100 / 400 = 0.25
p2 = 127 / 400 ≈ 0.3175

Now we can calculate the standard error:

Standard Error = sqrt((0.25 * (1 - 0.25) / 400) + (0.3175 * (1 - 0.3175) / 400))

After simplifying the equation, we get:
Standard Error = sqrt((0.25 * 0.75 / 400) + (0.3175 * 0.6825 / 400))

Standard Error ≈ sqrt(0.00046875 + 0.00054346875)
Standard Error ≈ sqrt(0.00101221875)
Standard Error ≈ 0.031817

Therefore, the approximate standard error for the statistic is 0.0318.

The correct answer is D. 0.0318.

To calculate the approximate standard error for the statistic, we can use the formula:

Standard Error = sqrt((p1 * (1 - p1))/ n1 + (p2 * (1 - p2))/ n2)

Here, p1 and p2 represent the proportions of successes in populations 1 and 2, respectively, and n1 and n2 represent the sample sizes.

First, let's calculate the proportions of successes for the two samples:

p1 = x1 / n1 = 100 / 400 = 0.25
p2 = x2 / n2 = 127 / 400 = 0.3175

Now, we can substitute these values into the formula:

Standard Error = sqrt((0.25 * (1 - 0.25))/ 400 + (0.3175 * (1 - 0.3175))/ 400)
= sqrt((0.1875)/ 400 + (0.2132)/ 400)
= sqrt(0.00046875 + 0.00053225)
= sqrt(0.001001)

Calculating the square root, we get:

Standard Error ≈ 0.031696

Therefore, the approximate standard error for the statistic is approximately 0.0318.

So, the correct option is D. 0.0318.