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?

Sample Size:
→ Generally, larger sample sizes result in smaller margins of error because larger samples provide more precise estimates of the population parameter.

Variability in Responses:
→ Higher variability in responses within a sample can lead to larger margins of error because it indicates less consistency in the population’s opinions or behaviors.

Given that the samples were selected from the same population and the margins of error were calculated using the same method, we can focus on the differences between the two samples.

Here’s how we can approach the solution:
Analyze Sample A and Sample B:
→ Examine the sample sizes of both samples.
→ Consider any differences in the variability of responses between the two samples.

Identify the Reason for the Difference in Margin of Error:
→ Select the option that aligns with the identified reason for the difference in margin of error.

Let’s say:
→ Sample A has a smaller sample size compared to Sample B.
→ Both samples exhibit similar variability in responses.

In this case, the most appropriate reason for the margin of error for sample A being greater than the margin of error for sample B would be the difference in sample sizes.

Therefore, the solution is "Sample A had a smaller sample size.

This option suggests that there might be missing data in sample A, which could potentially affect the accuracy of the estimation. However, the margin of error is primarily influenced by the sample size and the variability of responses, not by the number of missing votes

Sample A had a smaller number of votes that could not be recorded

This option implies that sample A had a higher proportion of votes in favor of the proposition compared to sample B. While the distribution of responses can impact the margin of error, it’s not directly related to why the margin of error for sample A is greater

Sample A had a higher percent of favorable responses

If sample A had a larger sample size, it would generally result in a smaller margin of error. Larger sample sizes provide more precise estimates of the population parameter, leading to a smaller margin of error. Therefore, this option is unlikely to be the reason for the margin of error for sample A being greater

Sample A had a larger sample size

This option aligns with the principles of sampling and margin of error calculations. A smaller sample size typically results in a larger margin of error because there is more uncertainty in estimating the population parameter with fewer data points. Therefore, if sample A had a smaller sample size compared to sample B, it would likely have a greater margin of error

Sample A had a smaller sample size

Option

D

Can you summarize what you’ve understood in the above steps?