Question

A research firm wishes to conduct a study of voters in Washington County who favor a certain state initiative. Previous studies reveal that 60% of voters favor the initiative. Researchers wish to conduct another survey in which the margin of error is no more than 3.0%. Assuming a 90% confidence level, answer the following questions. What is the positive critical value alpha/2? (round to 3 decimal places) What is the sample size needed to insure the desired margin of error = (rounded up to the nearest whole number)

Answer 1

For a 90% confidence level, the positive critical value (α/2) is typically around 1.645. The sample size needed to ensure a margin of error of no more than 3.0% is approximately 753, when rounded up to the nearest whole number.

Step-by-step explanation:

To determine the positive critical value (α/2) for a 90% confidence level, we would look up the value in a standard normal distribution table or use statistical software. However, since the exact number is not provided in the question, we cannot calculate the exact value but typically for a 90% confidence interval, the critical value is around 1.645. This would need to be rounded to three decimal places as per the question's requirements.

To ensure the margin of error is no more than 3.0%, we can use the formula for the sample size necessary to estimate a proportion:

n = (Z ² * p * (1 - p)) / E²,

where Z is the Z-score corresponding to the desired confidence level, p is the estimated proportion of the population that favors the initiative (0.60), and E is the desired margin of error (0.03).

Using the assumed Z-score for a 90% confidence level (1.645), the calculation would be as follows:

n = (1.645² * 0.60 * (1 - 0.60)) / 0.03²,

n ≈ 752.953.

We would round up to the nearest whole number, so the required sample size is 753.