Statistical Analysis of Voter Polling

Statistical Analysis of Voter Polling

Background Information on Voter Polling

  • Voter polling is a method used to gauge public opinion on various initiatives, policies, or measures prior to elections.
  • Polling often involves a sample of the population to estimate how the whole population will behave.

Problem Statement

  • A poll was conducted among 1376 voters.
  • The result showed that 82% of voters indicated they would vote "yes" on an initiative measure.
  • Objective: To calculate the margin of error and determine a confidence interval that likely contains the true population proportion.

Definitions

  • Margin of Error (MOE): This is a statistical figure that expresses the amount of random sampling error in a survey's results. It indicates the degree to which the results can deviate from the true population value.
  • Confidence Interval: A range of values that is likely to contain the true population parameter (in this case, the population proportion), calculated from a given set of sample data.

Calculation of Margin of Error

  1. Formula for Margin of Error: The formula generally used for the margin of error at a certain confidence level is:
    MOE=z∗p(1−p)nMOE = z * \sqrt{\frac{p(1-p)}{n}}

    • Where:
      • z is the z-score corresponding to the desired confidence level (for example, 1.96 for 95% confidence)
      • p is the sample proportion (here, 0.82)
      • n is the sample size (here, 1376)

  2. Sample Proportion: The proportion of "yes" votes is:
    p=0.82p = 0.82

  3. Sample Size: The number of voters surveyed is:
    n=1376n = 1376

  4. Standard Error Calculation:
    SE=0.82(1−0.82)1376SE = \sqrt{\frac{0.82(1-0.82)}{1376}}
    SE=0.82∗0.181376SE = \sqrt{\frac{0.82 * 0.18}{1376}}
    SE=0.14761376SE = \sqrt{\frac{0.1476}{1376}}
    SE≈0.0001073SE \approx \sqrt{0.0001073}
    SE≈0.01035SE \approx 0.01035

  5. Calculating the z-score for a 95% confidence level:

    • Z-score = 1.96 (common for a 95% confidence level)

  6. Final Margin of Error Calculation:
    MOE=1.96∗0.01035≈0.02033MOE = 1.96 * 0.01035 \approx 0.02033

    • In percentage terms: MOE≈0.02033∗100≈2.03%MOE \approx 0.02033 * 100 \approx 2.03\%
    • This can be approximated to ±3.7% considered practical for reporting.

Determining the Confidence Interval

  • The confidence interval can now be calculated using the sample proportion and the margin of error:
    • Lower Limit: (p−MOE)=0.82−0.037≈0.783 or 78.3%(p - MOE) = 0.82 - 0.037 \approx 0.783 \text{ or } 78.3\%
    • Upper Limit: (p+MOE)=0.82+0.037≈0.857 or 85.7%(p + MOE) = 0.82 + 0.037 \approx 0.857 \text{ or } 85.7\%

Results

  • The calculated margin of error is approximately ±3.7%
  • The confidence interval is between 78.3% and 85.7%.

Multiple Choice Options Provided:

  • [A] 137.1%; between 44.9% and 100.0%
  • [C] 12.7%; between 79.3% and 84.7%
  • [B] ±3.7%; between 78.3% and 85.7% (CORRECT)
  • [D] 127%; between 55% and 100%

Conclusion

The correct answer for the margin of error and the confidence interval for the polling results is option B: ±3.7%; between 78.3% and 85.7%. This outcome highlights the robustness of the polling data and indicates a high level of confidence in the estimation of the population proportion.