| Sample | Sample size | Percent who support the bill | Margin of error |
|---|---|---|---|
| C | 900 | ||
| D | 900 |
Two independent random samples of registered voters were selected from the same large city. The margins of error, each calculated at a confidence level, are shown. The samples were selected using the same method and have the same size.
Which of the following is the most appropriate reason that the margin of error for sample D is greater than the margin of error for sample C?
For a conceptual margin-of-error question, first compare the factors explicitly held constant, such as sample size and confidence level. Then focus on the estimated proportions: for equally sized samples at the same confidence level, proportions closest to have the greatest sampling variability and thus the largest margin of error. Do not confuse margin of error with bias, which concerns whether a sample may be systematically unrepresentative.
Hints
Compare what is the same and what is different
Identify which features are the same for both samples. Then determine which remaining feature could explain the different margins of error.
Rule out stated differences
The samples have equal sizes, and both margins of error use a confidence level. Eliminate answer choices that conflict with either fact.
Focus on the estimated percentages
Compare and . For a proportion, think about whether an estimate near or one near an extreme such as has greater sampling variability.
Desmos Guide
Use a variability graph as an optional verification
This question is fastest to answer by comparing the stated conditions. To verify the relationship in Desmos, enter y=x(1-x) with the restriction 0<=x<=1.
Compare the two sample proportions
Plot A=(0.18,0.18(1-0.18)) and B=(0.49,0.49(1-0.49)). Point B is higher on the graph, showing that a proportion near has greater variability than one near .
Step-by-step Explanation
Identify what is held constant
The samples have the same size, and both margins of error are calculated at the same confidence level. Therefore, neither sample size nor confidence level explains the difference in the margins of error.
Compare the estimated percentages
Sample C has an estimated support rate of , while sample D has an estimated support rate of . Sample D's estimated percentage is much closer to .
Use the variability of proportions
For samples of the same size at the same confidence level, estimated percentages near have the greatest sampling variability. Estimated percentages closer to or have less sampling variability.
Choose the explanation
Because is closer to than is, sample D has greater sampling variability and therefore a greater margin of error. The correct choice is The estimated percentage in sample D is closer to , where sampling variability for proportions is greatest.