Question

Consider taking a random sample from a population with \(p=0.25\) a. What is the standard error of \(\hat{p}\) for random samples of size \(400 ?\) b. Would the standard error of \(\hat{p}\) be smaller for random samples of size 200 or samples of size \(400 ?\) c. Does cutting the sample size in half from 400 to 200 double the standard error of \(\hat{p} ?\)

Short answer

a) The standard error of \(\hat{p}\) for random samples of size 400 is obtained by plugging \(p=0.25\) and \(n=400\) into the formula \[ SE =\sqrt{\frac{{p(1-p)}}{{n}} } \] ; (b) The standard error of \(\hat{p}\) would be smaller for samples of size 400 than for samples of size 200; (c) Cutting the sample size in half from 400 to 200 does not exactly double the standard error of \(\hat{p}\), it increases it by the factor of \(\sqrt{2}\), which is about 1.41 - smaller than doubling.

Step-by-step solution

Calculate the standard error for a sample size of 400

We need to provide the standard error (SE) for a sample of size 400. By using the formula above, SE can be calculated as: \[ SE_1 = \sqrt{\frac{{0.25*(1-0.25)}}{{400}} } \]

Compare the standard error for samples of size 200 and 400

The question now inquires if the standard error would be smaller for random samples of size 200 or size 400. Let's calculate the standard error for a sample size of 200 following the same formula: \[ SE_2 = \sqrt{\frac{{0.25*(1-0.25)}}{{200}} } \]Then, compare the value obtained with the SE computed for the sample of size 400.

Determine relationship of cutting sample size in half with the standard error

The exercise asks whether cutting the sample size in half from 400 to 200 doubles the standard error or not. For this, we need to compare the two standard errors obtained in steps 1 and 2.