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 proportion of all voters in the state who favor approval.

To construct a confidence interval for the true proportion of voters in the state who favor approval, we can use the formula:

Confidence Interval = Sample proportion ± Margin of Error

First, we need to find the sample proportion, which is calculated by dividing the number of voters who favor approval by the total number of voters in the survey:

Sample Proportion (p̂) = Number of voters who favor approval / Total number of voters in the survey

p̂ = 408 / 925 = 0.4416

Next, we need to calculate the margin of error, which depends on the desired confidence level and the sample size. The formula to calculate the margin of error is:

Margin of Error = Critical Value * Standard Error

The critical value is determined based on the desired confidence level. For a 90% confidence level, we need to find the critical value corresponding to a 5% (1 - 0.90) significance level.

Using a standard normal distribution table or a statistical calculator, we find that the critical value for a 90% confidence level is approximately 1.645.

The standard error is calculated as:

Standard Error = sqrt( (Sample Proportion * (1 - Sample Proportion)) / Sample Size)

Standard Error = sqrt( (0.4416 * (1 - 0.4416)) / 925)

Standard Error = sqrt( 0.2457 / 925)

Standard Error ≈ 0.01561

Now we can calculate the margin of error:

Margin of Error = 1.645 * 0.01561 ≈ 0.02571

Finally, we can construct the confidence interval:

Confidence Interval = Sample Proportion ± Margin of Error

Confidence Interval = 0.4416 ± 0.02571

Confidence Interval ≈ (0.4159, 0.4673)

Therefore, with 90% confidence, we estimate that the true proportion of all voters in the state who favor approval is between approximately 0.4159 and 0.4673.