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How large a sample size would be needed to estimate the percentage of wireless routers in San Francisco that use data encryption, with an error of + 2 percent and 95 percent confidence?

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Final answer:

To estimate the percentage of wireless routers using data encryption in San Francisco with a 2 percent margin of error and 95 percent confidence, one would need a sample size of 2401 routers.

Step-by-step explanation:

To estimate the percentage of wireless routers in San Francisco that use data encryption with an error of ± 2 percent and 95 percent confidence, we need to calculate the necessary sample size. The formula to calculate the sample size for estimating proportions is:

n = (Z² × p × (1 - p)) / E²

Where:

  • Z is the Z-score corresponding to the desired confidence level
  • p is the estimated proportion of the population
  • E is the margin of error

For a 95 percent confidence level, the Z-score is approximately 1.96. If we do not have a prior estimate of p, it is conservative to use 0.5 as this maximizes the product p(1 - p), thus giving the largest sample size. In this case:

n = (1.96² × 0.5 × 0.5) / 0.02²

Calculating this, we get:

n = (3.8416 × 0.25) / 0.0004

n = 0.9604 / 0.0004 = 2401

Therefore, a sample size of 2401 wireless routers would be needed for a margin of error of ± 2 percent with 95 percent confidence.

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