Political Polls Typically Sample Randomly From The US Popula ✓ Solved

Political Polls Typically Sample Randomly From The Us Population

Political polls typically sample randomly from the U.S population to investigate the percentage of voters who favor some candidate or issue. The number of people polled is usually on the order of 1000. Suppose that one such poll asks voters how they feel about the President’s handling of the crisis in the financial markets. The results show that 575 out of the 1280 people polled say they either “approve” or “strongly approve” of the President’s handling of this matter. Based on the sample referenced above, find a 95% confidence interval estimate for the proportion of the entire voter population who “approve” or “strongly approve” of the President’s handling of the crisis in the financial markets.

Now, here’s an interesting twist. If the same sample proportion was found in a sample twice as large—that is, 1150 out of 2560—how would this affect the confidence interval? Your post should reflect your understanding of the question posed. In addition to the computations you employed to arrive at your response, your post must contain comments regarding the rationale for the approach you utilized.

Paper For Above Instructions

Understanding political polls is vital in gauging public opinion on various issues and candidates. The scenario presented involves calculating a 95% confidence interval estimate for the proportion of the U.S. population that approves of the President's handling of the financial crisis, based on survey data. We will first calculate the confidence interval using the original sample of 1280 respondents and then analyze how this interval changes with a larger sample size of 2560 respondents.

Step 1: Calculating the Sample Proportion

To find the confidence interval, we start by determining the sample proportion (p̂) of voters who approve or strongly approve of the President's actions. This is calculated using the formula:

p̂ = x / n

Where:

  • x = number of voters who approve (575)
  • n = total number of respondents (1280)

Thus, the sample proportion is:

p̂ = 575 / 1280 = 0.4484

Step 2: Finding the Standard Error

The next step is to calculate the standard error (SE) of the sample proportion. The standard error is given by:

SE = sqrt((p̂(1 - p̂)) / n)

Substituting the values we have:

SE = sqrt((0.4484 * (1 - 0.4484)) / 1280)

SE = sqrt((0.4484 * 0.5516) / 1280)

SE = sqrt(0.2478 / 1280) = sqrt(0.0001930) ≈ 0.0139

Step 3: Calculating the Z-score for 95% Confidence Level

For a 95% confidence level, the Z-score (Z*) can be found using a standard Z-table, which is approximately 1.96.

Step 4: Calculating the Margin of Error

The margin of error (ME) can be calculated using the formula:

ME = Z * SE

Substituting the values:

ME = 1.96 * 0.0139 ≈ 0.0273

Step 5: Constructing the Confidence Interval

The confidence interval can be determined by the following:

CI = (p̂ - ME, p̂ + ME)

CI = (0.4484 - 0.0273, 0.4484 + 0.0273)

CI ≈ (0.4211, 0.4757)

This means we are 95% confident that the true proportion of the entire voter population that approves or strongly approves of the President’s handling of the crisis falls between approximately 42.11% and 47.57% based on this sample.

Effect of Increasing the Sample Size

Now, considering a sample size of 2560 respondents where the same proportion is found (1150 approving), we will repeat the calculations for the confidence interval.

Step 1: New Sample Proportion

For a sample size of 2560:

p̂ = 1150 / 2560 = 0.4484

Notice that the sample proportion remains the same.

Step 2: New Standard Error

New standard error:

SE = sqrt((0.4484 * (1 - 0.4484)) / 2560)

SE = sqrt((0.4474 * 0.5516) / 2560)

SE = sqrt(0.2478 / 2560) = sqrt(0.0000968) ≈ 0.0098

Step 3: Margin of Error with New Sample Size

New margin of error:

ME = 1.96 * 0.0098 ≈ 0.0192

Step 4: New Confidence Interval

The new confidence interval is:

CI = (0.4484 - 0.0192, 0.4484 + 0.0192)

CI ≈ (0.4292, 0.4676)

Comparing the two intervals:

  • Original Interval: (0.4211, 0.4757)
  • New Interval: (0.4292, 0.4676)

The narrowed confidence interval resulting from the larger sample size illustrates an important aspect of statistical sampling: larger sample sizes provide more precise estimates of population parameters, reducing the margin of error.

Conclusion

In conclusion, through statistical analysis, we found that the proportion of voters approving the President's handling of the financial crisis was estimated at about 44.84%. The confidence interval varies with sample size; a larger sample yields a narrower interval, enhancing the precision of our estimates. Therefore, it's critical for pollsters to consider sample size as a fundamental aspect when interpreting polling results.

References

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  • Brown, L. D., Cai, T. T., & DasGupta, A. (2001). Interval Estimation for a Binomial Proportion. Statistical Science, 16(2), 101-133.
  • Casella, G., & Berger, R. L. (2002). Statistical Inference. Cengage Learning.
  • Freedman, D. A., Pisani, R., & Purves, R. (2007). Statistics. W.W. Norton & Company.
  • Lehmann, E. L., & Casella, G. (2006). Theory of Point Estimation. Springer.
  • Newman, D. J. (2007). Principles of Statistical Reasoning. Emerald Group Publishing Limited.
  • Siegel, S., & Castellan, N. J. (1988). Nonparametric Statistics for the Behavioral Sciences. McGraw-Hill.
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