In A Current Election Campaign Psi Has Just Found That 220 R

In A Current Election Campaign Psi Has Just Found That 220 Registered

In a current election campaign, PSI has just found that 220 registered voters, out of 500 contacted, favor a particular candidate. a) What’s the 95% confidence interval estimate for the proportion of the population of registered voters that favor the candidate? b) Suppose that PSI would like a .99 probability that the sample proportion is within ± .03 of the population proportion. How large a sample size is needed to meet the required precision? (A previous sample of similar units yielded .44 for the sample proportion.)

Paper For Above instruction

Introduction

The process of estimating public opinion through sampling is fundamental in political polling, enabling campaigns to gauge voter preferences accurately while minimizing resource expenditure. This paper addresses two core statistical problems based on a recent poll conducted by PSI: calculating a confidence interval for the proportion of registered voters favoring a candidate and determining the necessary sample size to achieve a desired level of precision with a specified confidence level.

Part A: Confidence Interval Estimation for Population Proportion

Given that 220 out of 500 contacted voters favor the candidate, the sample proportion (p̂) is calculated as 0.44 (220/500). To estimate the true proportion (p) of all registered voters favoring the candidate with 95% confidence, we employ the formula for a confidence interval for a population proportion:

CI = p̂ ± z*√(p̂(1 - p̂)/n)

Where:

- p̂ = 0.44

- n = 500

- z* = 1.96 (the z-value corresponding to a 95% confidence level)

Calculating the standard error:

SE = √(0.44 * 0.56 / 500) = √(0.2464 / 500) ≈ √0.0004928 ≈ 0.0222

Margin of Error (ME):

ME = 1.96 * 0.0222 ≈ 0.0435

Therefore, the 95% confidence interval is:

(0.44 - 0.0435, 0.44 + 0.0435) = (0.3965, 0.4835)

Interpretation:

There is a 95% probability that the true proportion of registered voters favoring the candidate lies between approximately 39.65% and 48.35%. This interval provides a range acknowledging sampling variability, informing the campaign about the likely voter support levels in the broader population.

Part B: Determining Sample Size for Desired Precision

The campaign aims for a high confidence level of 99% that the sample proportion is within ±0.03 of the true population proportion. Given the previous sample proportion p̂ = 0.44, the required sample size (n) is calculated using the formula:

n = (z / E)^2 p̂(1 - p̂)

Where:

- z* = 2.576 (for 99% confidence)

- E = 0.03 (desired margin of error)

Plugging in the numbers:

n = (2.576 / 0.03)^2 0.44 0.56

n ≈ (85.867)^2 * 0.2464

n ≈ 7372.23 * 0.2464

n ≈ 1817.15

Rounding up, the necessary sample size is approximately 1818 voters to ensure with 99% confidence that the estimate is within ±3 percentage points of the true proportion.

Conclusion

The sampling analysis underscores the importance of confidence intervals and sample size calculations in political polling. The estimated confidence interval provides a range for voter support that the campaign can expect, while the sample size determination ensures that subsequent surveys can meet desired precision levels with high confidence. These statistical tools are essential in designing effective polling strategies that inform campaign decisions and resource allocations.

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