Question 12·Hard·Inference from Sample Statistics and Margin of Error
| Sample | Percent in favor | Margin of error |
|---|---|---|
| A | 52% | 4.2% |
| B | 48% | 1.6% |
The results of two random samples of votes for a proposition are shown above. The samples were selected from the same population, and the margins of error were calculated using the same method.
Which of the following is the most appropriate reason that the margin of error for sample A is greater than the margin of error for sample B?
For margin-of-error questions, immediately note whether samples come from the same population and use the same method; if so, focus on sample size as the key difference. Remember: larger sample → smaller margin of error, smaller sample → larger margin of error. Ignore tempting distractors about the exact percentage in favor or side issues like unrecorded votes, and instead directly match the margin-of-error sizes with the implied relative sample sizes.
Hints
Focus on what margin of error means
Margin of error tells you how far the sample result might be from the true population value. Think about what usually makes that uncertainty larger or smaller when you run a survey.
Use the "same population, same method" info
Because both samples are from the same population and use the same method, most factors are held constant. What important feature of a survey can still differ between samples?
Connect margin of error and the number of people surveyed
Imagine taking a survey of 10 people versus 1,000 people. In which case would you expect the result to be more precise, and how would the margin of error compare?
Compare A and B directly
Sample A’s margin of error is 4.2%, and sample B’s is 1.6%. Which sample must have been based on more data, and which answer choice says that about sample A?
Desmos Guide
Model the relationship between margin of error and sample size
In Desmos, type y = 1/sqrt(x) to model the general idea that margin of error (y) decreases as sample size (x) increases.
Observe how the graph behaves
Look at the graph: as x increases to the right (larger sample size), the y-values get smaller (smaller margin of error). As x gets smaller, y gets larger.
Connect the graph to Samples A and B
Use the pattern you see: the sample with the larger margin of error should correspond to a point with a smaller x-value (sample size), and the sample with the smaller margin of error should match a larger x-value. Decide what this means for sample A versus sample B.
Step-by-step Explanation
Interpret the table
From the table:
- Sample A: 52% in favor, margin of error 4.2%
- Sample B: 48% in favor, margin of error 1.6%
Both samples are from the same population, and the margins of error were found using the same method.
Recall what affects margin of error
For random samples from the same population using the same method, the biggest factor that changes the margin of error is sample size.
- Larger samples give more precise estimates and a smaller margin of error.
- Smaller samples give less precise estimates and a larger margin of error. The exact percent in favor (like 52% vs 48%) has only a small effect compared to sample size when both are near 50%.
Compare the two margins of error
Sample A’s margin of error (4.2%) is much larger than sample B’s (1.6%). Using the relationship from the previous step:
- The sample with the larger margin of error must be based on the smaller sample size.
- The sample with the smaller margin of error must be based on the larger sample size.
Match the reasoning to an answer choice
Since A has the larger margin of error, it must have the smaller sample size, and B must have the larger sample size. Therefore, the most appropriate reason is:
Sample A had a smaller sample size.