The Academy of Orthopedic Surgeons states that 80% of women wear shoes that are too small for their feet. A researcher wants to be 98% confident that this proportion is within 3% of the true proportion. How large a sample is necessary?
To determine the necessary sample size, we can use the formula for sample size calculation in proportion estimation. The formula is:
n = (Z^2 * p * (1-p)) / E^2
Where:
n: required sample size
Z: Z-value corresponding to the desired level of confidence (in this case, 98% confidence level)
p: estimated proportion (or the expected proportion of women wearing shoes that are too small)
E: margin of error (in this case, 3% proportion difference)
Let's plug in the given values and calculate the sample size:
Z = Z-value for 98% confidence level
Since we want to be 98% confident, the Z-value can be obtained from a standard normal distribution table or a statistical calculator. For a 98% confidence level, the Z-value is approximately 2.33.
p = estimated proportion = 0.8 (80%, as given in the statement)
E = margin of error = 0.03 (3%, as given in the statement)
Now, we can substitute the values into the formula:
n = (2.33^2 * 0.8 * (1-0.8)) / 0.03^2
n = (5.4289 * 0.16) / 0.0009
n ≈ 9643.22
Since the sample size should be a whole number, we round up to the nearest integer:
n = 9644
Therefore, a sample size of approximately 9644 individuals is necessary to be 98% confident that the proportion of women wearing shoes that are too small is within 3% of the true proportion.
To determine the sample size required to estimate the proportion of women wearing shoes that are too small for their feet with a 98% confidence level and a 3% margin of error, we can use the following formula:
n = (Z^2 * p * (1-p)) / E^2
Where:
n = sample size
Z = Z-score corresponding to the confidence level (98% confidence level corresponds to a Z-score of 2.33)
p = estimated proportion (If no estimate is available, assume p = 0.5 to get a conservative estimate for determining the sample size)
E = margin of error (0.03)
Using these values, let's calculate the required sample size:
n = (2.33^2 * 0.5 * (1-0.5)) / 0.03^2
n = (5.4289 * 0.25) / 0.0009
n = 1.357225 / 0.0009
n ≈ 1508.03
Since a sample size cannot be a fraction, we round up the result to the nearest whole number.
Therefore, a sample size of approximately 1509 is necessary to estimate the proportion of women wearing shoes that are too small for their feet with a 98% confidence level and a 3% margin of error.
Try this formula:
n = [(z-value)^2 * p * q]/E^2
...where n = sample size needed; use .8 for p and .2 for q (q = 1 - p). E = maximum error, which is .03 (3%) in the problem. Z-value is found using a z-table (for 98%, the value is 2.33). Also, ^2 means squared and * means to multiply.
I hope this will help get you started.