A researcher wishes to be 95% confident that her estimate of the population proportion of individuals who travel overseas is within 4% of the true proportion. Find the sample size necessary. In a prior study, a sample of 200 people showed that 80 traveled overseas last year.
To find the necessary sample size, we can use the formula for sample size calculation for estimating a population proportion. The formula is:
n = (Z^2 * p * q) / E^2
Where:
n = sample size
Z = Z-score corresponding to the desired confidence level (95% confidence level corresponds to Z = 1.96)
p = estimated proportion from previous study (80/200 = 0.4)
q = 1 - p
E = margin of error (in this case, 4% = 0.04)
Let's plug in the values and calculate the sample size:
n = (1.96^2 * 0.4 * 0.6) / 0.04^2
n = (3.8416 * 0.24) / 0.0016
n = 0.9226 / 0.0016
n ≈ 576.6
Rounding up to the nearest whole number, the sample size necessary is approximately 577.
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.
Formula to find sample size:
n = [(z-value)^2 * p * q]/E^2
... where n = sample size, z-value is found using a z-table for 95% confidence (which will be 1.96), p = 80/200, q = 1 - p, ^2 means squared, * means to multiply, and E = .04
Convert all fractions to decimals, then plug values into the formula and calculate n.
I hope this will help get you started.