How would I do this question. What sample size was needed to obtain an error range of 2% if the following statement was made. 75% of the workers support the proposed benefit package. The results are considered accurate to within + or- 2%, 18 times out of 20.
To determine the sample size needed to obtain an error range of 2% with a confidence level of 95% (18 times out of 20), you need to use the formula for calculating sample size for proportions.
Here's how you can solve this question step by step:
Step 1: Determine the confidence level and convert it into a Z-score.
The confidence level of 95% can be converted into a Z-score using a standard normal distribution table. In this case, the Z-score is 1.96.
Step 2: Calculate the maximum error or margin of error.
Since the desired error range is ±2%, the maximum error is 2%.
Step 3: Calculate the proportion.
The given statement is that 75% of the workers support the proposed benefit package. This proportion is denoted as p.
Step 4: Calculate the complement of the proportion.
Since you know the proportion p, you can calculate the complement of p, which is denoted as q. In this case, q = 1 - p = 1 - 0.75 = 0.25.
Step 5: Use the formula to calculate the required sample size.
The formula to calculate the required sample size for proportions is:
n = (Z^2 * p * q) / E^2
Substituting the values we have:
n = (1.96^2 * 0.75 * 0.25) / (0.02^2)
Simplifying the calculation:
n = (3.8416 * 0.75 * 0.25) / 0.0004
n = 0.7197 / 0.0004
n = 1799.25
Therefore, the sample size needed to obtain an error range of 2% with a confidence level of 95% is approximately 1799.25. Since sample sizes need to be whole numbers, you would round up to the nearest whole number. Hence, a sample size of 1800 would be needed.