A researcher conducted a study of the access speed of 45 hard drives and concluded that his maximum error of estimate was 28. If he were to conduct a second study to reduce the maximum error of estimate to 7, about how many hard drives should he include in his new sample?
Select one:
a. 720
b. 45
c. 90
d. 180
Please can someone explain me how to do the above problem.
Thanks!
To answer this question, we can use the formula for the sample size required to estimate a population mean with a maximum error of estimate:
n = (Z * σ / E)²
where:
n = sample size
Z = Z-score corresponding to the desired level of confidence (e.g., for a 95% confidence level, Z is approximately 1.96)
σ = standard deviation of the population
E = maximum error of estimate
In the first study, the researcher had a maximum error of estimate of 28. Therefore, the formula was:
n₁ = (Z * σ / 28)²
For the second study, the researcher wants to reduce the maximum error of estimate to 7. Therefore, the new formula becomes:
n₂ = (Z * σ / 7)²
To find the ratio of sample sizes between the two studies, we can divide the two equations:
n₂ / n₁ = [(Z * σ / 7) / (Z * σ / 28)]²
The σ cancels out, and we can simplify the equation:
n₂ / n₁ = (28 / 7)²
n₂ / n₁ = 16
So, to reduce the maximum error of estimate from 28 to 7, the researcher needs to increase the sample size by a factor of 16.
Since the first study included 45 hard drives, the researcher should include:
n₂ = n₁ * 16 = 45 * 16 = 720
Therefore, the researcher should include approximately 720 hard drives in the new sample.
The correct answer is (a) 720.