An article estimated the proportion of Google users who remember their passwords by writing then on a
piece of paper to be 0.383. The margin of error associated with this estimate is 0.035.
(a) Suppose that you would like to estimate the proportion of Google users who remember their
passwords by writing them on a piece of paper with a margin of error of 0.01. Would you want a
sample size that was smaller or larger than the one that produced a margin of error of 0.035?
(b) Using 0.383 as a preliminary estimate, what is the required sample size if you want to estimate the
proportion of Google users who remember their passwords by writing then on a piece of paper with a
margin of error of 0.01? (Round your answer up to the nearest integer.)
(c) How does the sample size in part (b) compare to the sample size that would result from using the
conservative value of 0.5? (Round your answer up to the nearest integer.)
(d) What sample size would you recommend? Justify your answer. (Round your sample size up to the
nearest integer.)
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(a) You would want a sample size that is larger than the one that produced a margin of error of 0.035. A smaller margin of error requires a larger sample size to achieve a more precise estimate.
(b) To estimate the required sample size, we can use the formula for sample size determination:
n = (z^2 * p * q) / E^2
where:
- n is the required sample size
- z is the z-value corresponding to the desired level of confidence (we'll assume a 95% confidence level, so z will be 1.96)
- p is the estimated proportion (0.383)
- q is 1 - p
- E is the desired margin of error (0.01)
Using these values, we can plug them into the formula:
n = (1.96^2 * 0.383 * (1-0.383)) / (0.01^2)
n = 884.872296
Rounding up to the nearest integer, the required sample size is 885.
(c) To compare the sample size using the conservative value of 0.5, we can plug in the values into the formula again:
n = (1.96^2 * 0.5 * (1-0.5)) / (0.01^2)
n = 9604
Rounding up to the nearest integer, the sample size using the conservative value of 0.5 is 9604.
(d) In this case, it would be recommended to use the sample size of 885. This sample size provides a more precise estimate with a smaller margin of error compared to the sample size of 9604. The conservative value of 0.5 assumes the largest possible variation, leading to a larger sample size requirement, but since we already have an estimate of 0.383, we can use it to reduce the required sample size.
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