To find the sample size necessary, we can use the formula:
n = (Z^2 * p * (1-p)) / E^2
Where:
n = required sample size
Z = Z-score for desired level of confidence (95% in this case)
p = estimated population proportion
E = margin of error (4% in this case)
First, let's find the Z-score for a 95% confidence level. The Z-score corresponds to the desired confidence level and can be obtained from a standard normal distribution table or calculated using statistical software. For a 95% confidence level, the Z-score is approximately 1.96.
Now, let's find the estimated population proportion (p). The researcher has a prior sample of 200 people, with 80 of them traveling overseas. Therefore, the estimated population proportion (p) is:
p = (number of individuals who traveled overseas) / (total sample size)
p = 80 / 200
p = 0.4
Finally, substitute the values into the formula:
n = (1.96^2 * 0.4 * (1-0.4)) / (0.04^2)
Simplifying the equation further:
n = (3.8416 * 0.24) / 0.0016
n = 57.984 / 0.0016
n ≈ 36,240
Therefore, the researcher will need a sample size of approximately 36,240 in order to be 95% confident that her estimate of the population proportion of individuals who travel overseas is within 4% of the true proportion.