Data from a representative sample were used to estimate that 33% of all computer users in a recent year
had tried to get on a Wi-Fi network that was not their own in order to save money. You decide to conduct a
survey to estimate this proportion for the current year. What is the required sample size if you want to
estimate this proportion with a margin of error of 0.05?
Calculate the required sample size first using 0.33 as a preliminary estimate of p. (Round your answer up to
the nearest integer.)
Calculate the required sample size using the conservative value of 0.5. (Round your answer up to the
nearest integer
How do the two sample sizes compare?
The sample size computed using the preliminary estimate is _____ the sample size
computed using the conservative value.
What sample size would you recommend for this study? (Round your sample size up to the nearest integer.)
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To calculate the required sample size using a preliminary estimate of p = 0.33, we use the formula:
n = (Z^2 * p * (1-p)) / E^2,
where Z is the z-score corresponding to the desired confidence level, p is the preliminary estimate of the proportion, and E is the margin of error.
Assuming a 95% confidence level, the z-score corresponding to this confidence level is approximately 1.96.
Using p = 0.33 and E = 0.05, we have:
n = (1.96^2 * 0.33 * (1-0.33)) / 0.05^2
n = (3.8416 * 0.33 * 0.67) / 0.0025
n = 4.9033344 / 0.0025
n ≈ 1961.33
Rounding up to the nearest integer, the required sample size using the preliminary estimate is 1962.
Now, let's calculate the required sample size using a conservative value of p = 0.5.
Using p = 0.5 and E = 0.05, we have:
n = (1.96^2 * 0.5 * (1-0.5)) / 0.05^2
n = (3.8416 * 0.5 * 0.5) / 0.0025
n = 0.9604 / 0.0025
n ≈ 384.16
Rounding up to the nearest integer, the required sample size using the conservative value is 385.
Comparing the two sample sizes, we can see that the sample size computed using the preliminary estimate is larger (1962) than the sample size computed using the conservative value (385).
For this study, it would be recommended to use the larger sample size of 1962 to ensure a more accurate estimate.
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