Question

Using a sample size that is 4 times as large will reduce the standard error of the sample proportion by one-half

Answer 1

The statement misrepresents the relationship between sample size and standard error, as the reduction is by one-half, not one-fourth, when the sample size is increased fourfold.

The statement about reducing the standard error of the sample proportion by one-half when using a sample size four times as large is not accurate. The relationship between sample size and standard error is not linear. In fact, the relationship follows a square root function.

When the sample size increases, the standard error of the sample proportion decreases, but not proportionally. It decreases by the square root of the sample size. So, if you have a sample size that is four times as large (quadrupled), the standard error will be reduced by the square root of four, which is two.

In mathematical terms, if
`\( SE_1 \)` is the standard error with the initial sample size, and
`\( SE_2 \)` is the standard error with a sample size four times as large, the relationship is given by:

`\[ SE_2 = (SE_1)/(√(4)) = (SE_1)/(2) \]`

So, the standard error is reduced by one-half, not one-fourth, when the sample size is quadrupled. It's important to understand the correct relationship between sample size and standard error to make informed decisions in statistical analysis.