A Political Candidate Has Asked You To Conduct A Poll

A Political Candidate Has Asked You To Conduct a Poll To Determine

A political candidate has asked you to conduct a poll to determine what percentage of people support him. If the candidate only wants a 0.5% margin of error at a 99% confidence level, what size of sample is needed? When finding the z-value, round it to four decimal places.

You want to obtain a sample to estimate a population proportion. At this point in time, you have no reasonable preliminary estimation for the population proportion. You would like to be 98% confident that your estimate is within 1% of the true population proportion. How large of a sample size is required?

A political candidate has asked you to conduct a poll to determine what percentage of people support her. If the candidate only wants a 5% margin of error at a 95% confidence level, what size of sample is needed? When finding the z-value, round it to four decimal places.

You want to obtain a sample to estimate a population proportion. Based on previous evidence, you believe the population proportion is approximately p = 0.14. You would like to be 99% confident that your estimate is within 5% of the true population proportion. How large of a sample size is required?

You want to obtain a sample to estimate a population proportion. Based on previous evidence, you believe the population proportion is approximately 81%. You would like to be 99% confident that your estimate is within 3.5% of the true population proportion. How large of a sample size is required?

You want to obtain a sample to estimate a population mean. Based on previous evidence, you believe the population standard deviation is approximately σ = 59.3. You would like to be 99% confident that your estimate is within 5 of the true population mean. How large of a sample size is required?

You want to obtain a sample to estimate how much parents spend on their kids' birthday parties. Based on previous study, you believe the population standard deviation is approximately σ = 60.4 dollars. You would like to be 99% confident that your estimate is within 2.5 dollars of average spending on the birthday parties. How many parents do you have to sample?

Paper For Above instruction

In the realm of statistical analysis, determining the appropriate sample size is fundamental to generating reliable estimates about a population. Whether estimating proportions or means, selecting an adequate sample ensures that the results are both accurate and reflective of the population characteristics. This paper explores various scenarios where sample size calculations are essential, emphasizing the importance of confidence levels, margins of error, and prior knowledge about population parameters.

Introduction

Sample size determination is a critical component in designing surveys and polls. It balances the need for statistical precision with logistical and resource constraints. The fundamental formulas vary depending on whether researchers are estimating population proportions or means. Accurate calculation of sample size hinges on understanding the desired confidence level, margin of error, the population standard deviation for means, and prior estimates of population parameters.

Sample Size for Estimating Population Proportions

When estimating a population proportion (p), the sample size (n) can be calculated using the formula:

n = (Z²  p  (1 - p)) / E²

where Z is the z-value associated with the confidence level, p is the estimated proportion, and E is the margin of error.

If there is no preliminary estimate of p, a conservative approach assumes p = 0.5, which maximizes the product p*(1-p).

Scenario 1: Support for a Candidate with a 0.5% Margin of Error at 99% Confidence

The z-value for a 99% confidence level is obtained from standard normal tables: approximately 2.5760 when rounded to four decimal places. Assuming no prior estimate for p, p=0.5. Plugging into the formula:

n = (2.5760)^2  0.5  0.5 / (0.005)^2 = (6.635) * 0.25 / 0.000025 = 1.65875 / 0.000025 = 66,350

Therefore, approximately 66,350 respondents are needed to achieve the specified precision.

Scenario 2: Estimating a Population Proportion with No Prior Estimate, 98% Confidence, 1% Margin

Here, p is unknown, so we use p=0.5. The z-value for 98% confidence is approximately 2.3263. The sample size is:

n = (2.3263)^2  0.5  0.5 / (0.01)^2 = (5.411) * 0.25 / 0.0001 = 1.35275 / 0.0001 = 13,527

Thus, a sample of about 13,527 is required.

Scenario 3: Support for Candidate with a 5% Margin at 95% Confidence

The z-value for 95% confidence is approximately 1.9594. Using p=0.5 to maximize the sample size:

n = (1.9594)^2  0.5  0.5 / (0.05)^2 = (3.839) * 0.25 / 0.0025 = 0.9597 / 0.0025 = 383.88

Rounding up, approximately 384 respondents are needed.

Scenario 4: Prior Estimate p=0.14, 99% Confidence, 5% Margin

The z-value for 99% confidence is 2.5760. Using the estimated p:

n = (2.5760)^2  0.14  0.86 / (0.05)^2 = 6.635 * 0.1204 / 0.0025 ≈ 0.798 / 0.0025 ≈ 319.2

Approximately 320 samples are necessary.

Scenario 5: Prior Estimate p=0.81, 99% Confidence, 3.5% Margin

The z-value remains 2.5760. Calculating:

n = (2.5760)^2  0.81  0.19 / (0.035)^2 = 6.635 * 0.1539 / 0.001225 ≈ 1.022 / 0.001225 ≈ 834

About 834 respondents are required.

Estimating a Population Mean with Known Standard Deviation

For estimating a mean (μ) where the population standard deviation (σ) is known, the sample size formula is:

n = (Z * σ / E)^2

Scenario 6: Mean Estimate with σ=59.3, 99% Confidence, Margin of 5

The z-value for 99% confidence is 2.5760. Plugging in:

n = (2.5760  59.3 / 5)^2 = (2.5760  11.86)^2 ≈ (30.55)^2 ≈ 933.3

Thus, about 934 samples are needed.

Scenario 7: Spending on Kids' Birthday Parties with σ=60.4, 99% Confidence, Margin of 2.5

Applying the formula:

n = (2.5760  60.4 / 2.5)^2 = (2.5760  24.16)^2 ≈ (62.33)^2 ≈ 3887.3

Approximately 3,888 parents should be sampled.

Conclusion

Determining the right sample size is a vital step in survey design, impacting the accuracy and reliability of the conclusions drawn. The calculations vary based on the type of parameter estimated, prior information available, confidence level, and acceptable margin of error. Proper application of these formulas ensures the collection of sufficient data to inform political campaigns, market research, or social studies effectively. Researchers must balance statistical rigor with practical constraints to optimize their survey design.

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