A survey of 925 voters in one state reveals that 408 favor approval of an issue before the legislature. Construct the 90% confidence interval for the true population of all voters in the state who favor approval.
To construct a confidence interval for the proportion of voters in the state who favor approval, we can use the formula:
CI = p̂ ± Z * sqrt((p̂(1-p̂))/n)
where:
- CI is the confidence interval
- p̂ is the sample proportion (favoring approval), which is calculated as p̂ = 408/925
- n is the sample size (925)
- Z is the critical value corresponding to the desired confidence level (90% confidence level corresponds to a Z-value of 1.645)
Let's calculate the confidence interval step by step:
1. Calculate the sample proportion:
p̂ = 408/925 ≈ 0.441
2. Calculate the standard error:
SE = sqrt((p̂(1-p̂))/n)
= sqrt((0.441(1-0.441))/925)
≈ 0.017
3. Calculate the margin of error:
ME = Z * SE
= 1.645 * 0.017
≈ 0.028
4. Calculate the lower and upper bounds of the confidence interval:
Lower bound = p̂ - ME
= 0.441 - 0.028
≈ 0.413
Upper bound = p̂ + ME
= 0.441 + 0.028
≈ 0.469
Therefore, the 90% confidence interval for the true population proportion of voters in the state who favor approval is approximately 0.413 to 0.469.
To construct the confidence interval, we can use the formula:
CI = p̂ ± z * √(p̂(1 - p̂)/n)
Where:
- CI is the confidence interval
- p̂ is the sample proportion (favor approval) = 408/925 = 0.441
- z is the z-score corresponding to the desired confidence level. For a 90% confidence interval, the z-score is approximately 1.645 (consult a z-table for the exact value)
- n is the sample size = 925
Plugging in the values, we can calculate the confidence interval as follows:
CI = 0.441 ± 1.645 * √(0.441 * (1 - 0.441)/925)
CI = 0.441 ± 1.645 * √(0.441 * 0.559/925)
CI = 0.441 ± 1.645 * √(0.2463/925)
CI = 0.441 ± 1.645 * √0.000266702
CI = 0.441 ± 1.645 * 0.016331
CI = 0.441 ± 0.026847
Therefore, the 90% confidence interval for the true population proportion of voters who favor approval is (0.414, 0.468).