Assume you want to estimate with the proportion of students who commute less than 5 miles to work within 2%, what sample size would you need?
Try this formula:
n = [(z-value)^2 * p * q]/E^2
Note: n = sample size needed; .5 for p and .5 for q are used if no value is stated in the problem. E = maximum error, which is .02; Z-value is found using a z-table (1.96 for 95% confidence as an example).
I'll let you take it from here.
1-.02=2.33=z-value
plug into the equation
[(2.33)^2*.5*.5]/(.02)^2
3393.06
To estimate the proportion of students who commute less than 5 miles to work within a certain margin of error, we need to calculate the sample size required. Here's how you can do it:
1. Determine the level of confidence: Typically, a 95% confidence level is used, which means that there is a 95% chance that the true population proportion falls within the estimated range.
2. Identify the estimated proportion: You need to have an estimate of the proportion of students who commute less than 5 miles to work. If you don't have an estimate, you can use a reasonable guess based on previous studies or conduct a preliminary survey to get an initial estimate.
3. Determine the desired margin of error: In this case, a margin of error of 2% is given. The margin of error represents the maximum amount of error allowed in the estimated proportion.
4. Use the formula: The sample size needed can be calculated using the following formula:
n = (Z^2 * p * (1-p)) / E^2
where:
- n is the required sample size
- Z is the z-score corresponding to the desired confidence level (e.g., for a 95% confidence level, Z = 1.96)
- p is the estimated proportion
- E is the desired margin of error
5. Plug in the values and calculate: Substitute the values into the formula to find the sample size.
For example, if the estimated proportion (p) is 0.50 (50%), the desired margin of error (E) is 0.02 (2%), and the z-score (Z) is 1.96, the calculation becomes:
n = (1.96^2 * 0.50 * (1-0.50)) / 0.02^2
Solving this equation will give you the required sample size.
By following these steps and evaluating the formula, you can calculate the sample size needed to estimate the proportion of students who commute less than 5 miles to work within a 2% margin of error.