Estimate Sample Size In Each Case At 95% Confidence

In Each Case Estimate The Sample Sizea Confidence Level 95

In each case estimate the sample size: (a) Confidence level = 95%, maximum error of the estimate = 2.45%, sample proportion known to be around 55%; estimating a population proportion; (b) Confidence level = 99%, maximum error of the estimate = 3.2, population standard deviation = 8.45; estimating a population mean; (c) Confidence level = 95%, maximum error of the estimate = 3.2, population standard deviation = 8.45; estimating a population mean; (d) Confidence level = 95%, maximum error of the estimate = 4.55%, no knowledge about the proportion; estimating a population proportion.

Paper For Above instruction

Determining the appropriate sample size is a fundamental aspect of statistical analysis, ensuring that estimates derived from samples are as accurate and reliable as possible while maintaining efficiency in resource allocation. The calculation of sample size varies depending on the type of parameter being estimated—population proportion or population mean—and depends on factors such as the desired confidence level, acceptable margin of error, and the available knowledge about the population parameter or variability. This paper systematically addresses each of the four cases provided, applying relevant formulas and principles to estimate the required sample sizes.

Case (a): Estimating a Population Proportion with Known Sample Proportion

In the first scenario, the goal is to estimate a population proportion with a 95% confidence level and a maximum error of 2.45%. The known sample proportion is approximately 55% (or 0.55). The formula to calculate the sample size (n) for a proportion is derived from the confidence interval for a proportion:

n = (Z_α/2)² * p̂(1 - p̂) / E²

Where:

• Z_α/2 is the z-value corresponding to the desired confidence level (for 95%, Z_0.975 ≈ 1.96),
• p̂ is the estimated proportion (0.55),
• E is the maximum error tolerated (0.0245).

Substituting the values:

n = (1.96)² 0.55 (1 - 0.55) / (0.0245)² ≈ 3.8416 0.55 0.45 / 0.00060025 ≈ 3.8416 * 0.2475 / 0.00060025 ≈ 0.951777 / 0.00060025 ≈ 1585.68

Thus, a sample size of approximately 1,586 respondents is required to estimate the population proportion with the specified confidence and precision.

Case (b): Estimating a Population Mean with Known Standard Deviation at 99% Confidence

In the second case, the objective is to estimate a population mean with a Z-value corresponding to a 99% confidence level, an acceptable error margin of 3.2, and a known population standard deviation of 8.45. The formula used is:

n = (Z_α/2)² * σ² / E²

Where:

• Z_0.995 ≈ 2.576 (for 99% confidence),
• σ = 8.45,
• E = 3.2.

Calculating:

n = (2.576)² (8.45)² / (3.2)² ≈ 6.635 71.4025 / 10.24 ≈ 473.964 / 10.24 ≈ 46.24

Rounding up, a sample size of 47 respondents would be sufficient to estimate the population mean within the specified margin of error with 99% confidence.

Case (c): Estimating a Population Mean with Known Standard Deviation at 95% Confidence

The third scenario is similar to the second, but with a confidence level of 95%. The value of Z_α/2 for 95% confidence is approximately 1.96. Reapplying the same formula:

n = (Z_α/2)² * σ² / E²

with the same σ and E:

n = (1.96)² (8.45)² / (3.2)² ≈ 3.8416 71.4025 / 10.24 ≈ 274.427 / 10.24 ≈ 26.8

Therefore, a sample size of approximately 27 respondents suffices for this estimate at 95% confidence.

Case (d): Estimating a Population Proportion without Known Proportion

In the final case, the goal is to estimate a population proportion with 95% confidence, a maximum error of 4.55%, but without prior knowledge of the proportion. When the true proportion p is unknown, the conservative approach is to assume p̂ = 0.5, which maximizes the product p̂(1 - p̂), leading to the largest required sample size:

n = (Z_α/2)² * p̂(1 - p̂) / E²

Using Z_0.975 ≈ 1.96, p̂ = 0.5, E = 0.0455:

n = (1.96)² 0.5 0.5 / (0.0455)² ≈ 3.8416 * 0.25 / 0.002070 ≈ 0.9604 / 0.002070 ≈ 464.17

Therefore, approximately 465 respondents are needed to estimate the proportion within the specified margin of error at 95% confidence when no prior estimate is available.

Conclusion

Accurate estimation of sample sizes tailored to specific parameters and constraints is crucial for effective statistical inference. The calculations demonstrate how varying the confidence level, margin of error, and prior knowledge influence the sample requirements. In practice, these estimations guide researchers in designing studies that balance precision, resource constraints, and statistical validity.

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