Question

An education fundraiser has been set up to collect donations for the most at-risk schools across the state of New York. A random sample of 500 people shows that 33% of the 250 who were contacted by telephone actually made contributions, compared with only 22% of the 250 who received email requests. Which of the formulas calculates the 99% confidence interval for the difference in the proportions of people who make donations when contacted by telephone versus those contacted by email?
Its either:
start quantity 0.33 minus 0.22 end quantity plus or minus 2.576 times the square root of start quantity 0.33 times 0.67 all over 500 plus 0.22 times 0.78 all over 500 end quantity
start quantity 0.33 minus 0.22 end quantity plus or minus 2.576 times the square root of start quantity 0.33 times 0.67 all over 250 plus 0.22 times 0.78 all over 250 end quantity

Answer 1

Using the z-distribution, as we are working with a proportion, the 99% confidence interval for the difference in the proportions is:

$(0.33 - 0.22) \pm 2.576\sqrt{\left(\sqrt{\frac{0.33(0.67)}{250}}\right)^2 + \left(\sqrt{\frac{0.22(0.78)}{250}}\right)^2}$

What is the mean and the standard error of the distribution of differences?

For each sample, we have that:

$p_T = 0.33, s_T = \sqrt{\frac{0.33(0.67)}{250}}$

$p_E = 0.22, s_E = \sqrt{\frac{0.22(0.78)}{250}}$

For the distribution of differences, the mean is the subtraction of the means, hence:

$p = p_T - p_E = 0.33 - 0.22$

The standard error is the square root of the sum of the variances, hence:

$s = \sqrt{s_T^2 + s_E^2} = \sqrt{\left(\sqrt{\frac{0.33(0.67)}{250}}\right)^2 + \left(\sqrt{\frac{0.22(0.78)}{250}}\right)^2}$

What is the confidence interval?

It is given by:

$p \pm zs$

99% confidence level, hence$\alpha = 0.99$, z is the value of Z that has a p-value of $\frac{1+0.99}{2} = 0.995$, so the critical value is z = 2.576.

Then, the interval is:

$(0.33 - 0.22) \pm 2.576\sqrt{\left(\sqrt{\frac{0.33(0.67)}{250}}\right)^2 + \left(\sqrt{\frac{0.22(0.78)}{250}}\right)^2}$

To learn more about the z-distribution, you can check https://brainly.com/question/25890103