A researcher wishes to be 95% confident that her estimate of the true proportion of individuals who travel overseas is within 4% if the true proportion. Find the sample necessary if in a prior study, a sample of 200 people showed that 40 traveled overseas last year. If no estimate of the same proportion is available, how large should a sample be????
show your work please so i can understand :P
(0.2)(0.8)(1.96/0.04)^2 =384.16 which is rounded to 385 to the next whole number
To find the necessary sample size, we can use the formula for sample size estimation in a proportion:
n = (Z^2 * p * (1-p)) / E^2
Where:
n = sample size
Z = Z-value for the desired level of confidence
p = estimated proportion
E = margin of error
Given information:
Desired level of confidence: 95% (implying a Z-value of 1.96 for a two-tailed test)
Margin of error: 4% (0.04)
Since there is no estimate of the same proportion available, we can use the sample proportion from the prior study as an estimate. In the prior study, out of a sample of 200 people, 40 traveled overseas (proportion = 40/200 = 0.2).
Now we can substitute the values into the formula:
n = (1.96^2 * 0.2 * (1-0.2)) / 0.04^2
Calculating the result:
n = (3.8416 * 0.2 * 0.8) / 0.0016
n = 0.61472 / 0.0016
n ≈ 384.2
Therefore, a sample size of approximately 384 people would be necessary to estimate the true proportion of individuals who travel overseas with a 95% confidence level and a margin of error of 4%.
To find the sample size required, we can use the formula for calculating the sample size for estimating a proportion. The formula is:
n = (Z^2 * p * (1-p)) / E^2
Where:
n = sample size
Z = Z-value for the desired level of confidence
p = estimated proportion from a prior study
E = desired margin of error
Given:
Confidence level = 95%, which corresponds to a Z-value of 1.96 (approximately)
Margin of error = 4% or 0.04
Estimated proportion from the prior study = 40/200 = 0.2
Using the given values in the formula:
n = (1.96^2 * 0.2 * (1 - 0.2)) / (0.04^2)
n = (3.8416 * 0.16) / 0.0016
n = 0.614656 / 0.0016
n ≈ 384.16
Rounding up to the nearest whole number, the researcher needs a sample size of 385 in order to be 95% confident that her estimate of the true proportion of individuals who travel overseas will be within 4% of the true proportion.
Therefore, the sample should be at least 385 individuals.
Try this formula:
n = [(z-value)^2 * p * q]/E^2
Note: n = sample size needed; use 40/200 for p (convert to a decimal); use 1 - p for the value of q. E = maximum error, which is .04 for 4%. Z-value is found using a z-table (for 95% confidence). Note: ^2 means squared and * means to multiply.
I'll let you take it from here.