A political consultant wants to sample residents of a county to estimate what proportion of all the county's residents support a certain candidate. They will use the sample data to construct a one-sample $z$ interval for the proportion, and they are considering two sample sizes: a large sample of $n=900$ for more accurate results, or a small sample of $n=100$ to save time and money.

Assuming the sample proportion is the same in each sample, what is true about the margins of error from these two samples?

The margin of error from the smaller sample will be about select ratio the margin of error from the larger sample.

Question

A political consultant wants to sample residents of a county to estimate what proportion of all the county's residents support a certain candidate. They will use the sample data to construct a one-sample $z$ interval for the proportion, and they are considering two sample sizes: a large sample of $n=900$ for more accurate results, or a small sample of $n=100$ to save time and money.

Assuming the sample proportion is the same in each sample, what is true about the margins of error from these two samples?

The margin of error from the smaller sample will be about select ratio the margin of error from the larger sample.

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ Identify the formula for the margin of error in a one-sample $z$ interval for the proportion. ‖ ‖ step 2 ⋮ The margin of error (ME) is calculated using the formula $ME = z \cdot \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$, where $z$ is the $z$-score corresponding to the desired confidence level, $\hat{p}$ is the sample proportion, and $n$ is the sample size. ‖ ‖ step 3 ⋮ Since the sample proportion $\hat{p}$ is the same for both samples, and the $z$-score is constant for a given confidence level, the margin of error is inversely proportional to the square root of the sample size $n$. ‖ ‖ step 4 ⋮ Calculate the ratio of the margins of error for the two sample sizes: $\frac{ME_{small}}{ME_{large}} = \frac{z \cdot \sqrt{\frac{\hat{p}(1 - \hat{p})}{100}}}{z \cdot \sqrt{\frac{\hat{p}(1 - \hat{p})}{900}}}$. ‖ ‖ step 5 ⋮ Simplify the ratio by canceling out common factors: $\frac{ME_{small}}{ME_{large}} = \frac{\sqrt{\frac{1}{100}}}{\sqrt{\frac{1}{900}}} = \frac{\sqrt{9}}{\sqrt{1}} = 3$. ‖ ∻Answer∻ ⚹ The margin of error from the smaller sample will be about 3 times the margin of error from the larger sample. ⚹ ∻Key Concept∻ ⚹Margin of Error and Sample Size Relationship⚹ ∻Explanation∻ ⚹The margin of error for a proportion is inversely proportional to the square root of the sample size, which means that when the sample size is multiplied by a factor, the margin of error is divided by the square root of that factor.⚹◊What is the formula to calculate the margin of error in a statistical sample?⍭ Generate me a similar question◊

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