The Academy of Orthopedic Surgeons states that 90% of US women wear shoes that are too small for their feet. A researcher wants to be 95% confident that this proportion is within 2% of the true proportion. How large a sample is necessary?
To determine the sample size required to estimate the proportion with a given level of confidence and a desired margin of error, we can use the formula for sample size for proportions.
The formula for sample size for proportions is given by:
n = (Z^2 * p * (1-p)) / E^2
where:
n = sample size
Z = Z-value corresponding to the desired confidence level
p = estimated proportion
E = maximum allowable margin of error
In this case, the researcher wants to be 95% confident, which corresponds to a Z-value of approximately 1.96 (for a two-tailed test). The researcher also wants the proportion estimate to have a maximum margin of error of 2%, which is 0.02.
Given that the Academy of Orthopedic Surgeons states that 90% of US women wear shoes that are too small for their feet, we can use this as the estimated proportion (p = 0.9).
Now we can calculate the sample size using the formula:
n = (1.96^2 * 0.9 * (1-0.9)) / (0.02^2)
Simplifying the equation:
n = (3.8416 * 0.9 * 0.1) / 0.0004
n = 0.346656 / 0.0004
n = 866.64
Rounding up to the nearest whole number, the researcher would need a sample size of approximately 867 to be 95% confident that the proportion of US women wearing shoes that are too small is within 2% of the true proportion.