Question 1 In A Poll Of 400 Voters In A Campaign

Question 1 In a poll of 400 voters in a campaign to E

Question 1 In a poll of 400 voters in a campaign to E

Develop a 99% confidence interval estimate for the proportion of all the voters who opposed the container control bill.

Paper For Above instruction

The estimation of population proportions through confidence intervals is a fundamental aspect of inferential statistics, providing a range within which the true proportion is likely to lie with a certain level of confidence. In this analysis, we examine a poll conducted among 400 voters concerning a campaign to eliminate non-returnable beverage containers. Out of these voters, 230 opposed the bill, resulting in an observed sample proportion that serves as the basis for constructing a 99% confidence interval estimate for the true proportion of all voters who oppose the bill.

First, it is important to determine the sample proportion (p̂). This is calculated by dividing the number of voters opposed by the total number of voters surveyed:

p̂ = 230 / 400 = 0.575

To construct the confidence interval, we use the formula for a proportion's confidence interval:

p̂ ± Zα/2 * √(p̂(1 - p̂) / n)

Where:

  • p̂ = 0.575, the sample proportion;
  • n = 400, the sample size;
  • Zα/2 = Z-value corresponding to the desired confidence level (99%).

Looking up the Z-value for a 99% confidence level (α = 0.01), and a two-tailed test, we find:

Zα/2 ≈ 2.576

Next, we compute the standard error (SE):

SE = √(p̂(1 - p̂) / n) = √(0.575 * 0.425 / 400) ≈ √(0.244 / 400) ≈ √0.00061 ≈ 0.0247

Now, the margin of error (ME):

ME = Zα/2 SE = 2.576 0.0247 ≈ 0.0637

Finally, the 99% confidence interval estimate for the true proportion p is:

[0.575 - 0.0637, 0.575 + 0.0637] = [0.5113, 0.6387]

This interval suggests that we are 99% confident the true proportion of all voters opposed to the container control bill lies between approximately 51.13% and 63.87%.

Understanding confidence intervals enables policymakers and campaign strategists to gauge public opinion reliably, which is essential for decision-making and policy formulation. The select confidence level reflects a high degree of certainty, beneficial in significant political or social campaigns.

References

  • Agresti, A., & Coull, B. A. (1998). Approximate is better than "exact" for interval estimation of binomial proportions. The American Statistician, 52(2), 119-126.
  • Newcombe, R. G. (1998). Two-sided confidence intervals for the single proportion: comparison of seven methods. Statistics in Medicine, 17(8), 857-872.
  • Wooldridge, J. M. (2010). Econometric Analysis of Cross Section and Panel Data. MIT Press.
  • Casella, G., & Berger, R. L. (2002). Statistical Inference. Duxbury Press.
  • Moore, D. S., McCabe, G. P., & Craig, B. A. (2012). Introduction to the Practice of Statistics. W. H. Freeman.
  • Cochran, W. G. (1977). Sampling Techniques. John Wiley & Sons.
  • Salzberg, S. (2018). Confidence intervals in practice: Interpretation and misinterpretation. Journal of Statistical Education, 26(2).
  • McClave, J. T., & Sincich, T. (2018). Statistics. Pearson.
  • Efron, B., & Tibshirani, R. J. (1993). An Introduction to the Bootstrap. Chapman & Hall/CRC.
  • Rothman, K. J., & Greenland, S. (1998). Modern Epidemiology. Lippincott Williams & Wilkins.

Related Assignments