A Researcher Wishes To Estimate With 90% Confidence The Prop

A Researcher Wishes To Estimate With 90 Confidence The Proportion O

A researcher wishes to estimate, with 90% confidence, the proportion of adults who have high-speed internet access. Her estimate must be accurate within 5% of the true proportion. a) Find the minimum sample size needed, using a prior study that found that 36% of the respondents said they have high-speed internet access. b) No preliminary estimate is available. Find the minimum sample size needed. (round answers to the nearest whole number)

Paper For Above instruction

The task is to determine the minimum sample size required to estimate the proportion of adults with high-speed internet access at a 90% confidence level, with a margin of error of 5%. The problem is approached in two scenarios: one using a prior estimate of the proportion and the other assuming no prior estimate is available. We’ll explore both cases, apply relevant statistical formulas, and provide detailed calculations to arrive at the required sample sizes.

Understanding the Context and Objectives

Estimating proportions with specified confidence and precision is a common problem in survey sampling. The researcher’s goal is to determine the smallest sample size that ensures the estimated proportion falls within ±5% of the true proportion with 90% confidence. The margin of error (E) is 0.05, and the confidence level corresponds to a z-score of approximately 1.645, based on the standard normal distribution. The key variables are:

- p̂: the estimated proportion

- p: the true proportion (unknown without prior data)

- E: margin of error (0.05)

- Z: z-score for 90% confidence (1.645)

Part A: Using a Prior Estimate of Proportion

When prior data suggests that 36% of respondents have high-speed internet access, this estimate (p̂ = 0.36) can be used directly to calculate the required sample size. The formula for the margin of error in estimating a proportion is:

n = (Z²  p̂  (1 - p̂)) / E²

Where:

- Z = 1.645 (for 90% confidence)

- p̂ = 0.36

- E = 0.05

Calculating the sample size

Plugging in the values:

n = (1.645²  0.36  (1 - 0.36)) / 0.05²

Calculating step-by-step:

  • 1.645² ≈ 2.705
  • p̂(1 - p̂) = 0.36 0.64 = 0.2304
  • Numerator: 2.705 * 0.2304 ≈ 0.623
  • Denominator: 0.05² = 0.0025

Therefore:

n = 0.623 / 0.0025 ≈ 249.2

Rounded to the nearest whole number: 249. Hence, the minimum sample size needed is 249 respondents.

Part B: No Prior Estimate of Proportion

In the absence of prior information about the proportion, the most conservative estimate is to assume p̂ = 0.5. This value maximizes the product p̂*(1 - p̂), resulting in the largest required sample size, ensuring adequate precision regardless of the actual proportion.

Calculating the sample size without prior estimate

n = (Z²  p̂  (1 - p̂)) / E²
Plugging in the values:
  • Z = 1.645
  • p̂ = 0.5
  • 1 - p̂ = 0.5
n = (1.645²  0.5  0.5) / 0.05²
Step-by-step calculation:
  • 1.645² ≈ 2.705
  • p̂(1 - p̂) = 0.5 0.5 = 0.25
  • Numerator: 2.705 * 0.25 ≈ 0.676
  • Denominator: 0.0025
n = 0.676 / 0.0025 ≈ 270.4

Rounded to the nearest whole number: 270. Thus, when no prior estimate exists, the required sample size is 270 respondents.

Discussion and Implications

The calculations illustrate how prior data can influence sample size determination significantly. Utilizing the previous estimate (36%) reduces the sample size from 270 to 249, potentially saving resources. However, if such prior data is unreliable or unavailable, defaulting to the conservative estimate ensures robustness of the survey results.

Conclusion

In summary, to estimate the proportion of adults with high-speed internet access at 90% confidence and a ±5% margin of error:

- With a prior estimate of 36%, a sample size of 249 is needed.

- Without prior data, assuming p̂=0.5, a sample size of 270 is necessary.

References

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  • Pakula, B., & Kaiser, H. M. (2013). Sample Size Calculations for Proportions. Wiley StatsRef: Statistics Reference Online.
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