To construct a confidence interval for the difference in proportions $(p_1 - p_2)$, we will follow these steps:

Step 1: Identify the Given Information

• Confidence Level: 90%
• The formula for the confidence interval for the difference between two population proportions is given by:

$(p_1 - p_2) \pm Z_{\alpha/2} \cdot \sqrt{\frac{p_1(1 - p_1)}{n_1} + \frac{p_2(1 - p_2)}{n_2}}$

Where:

• $p_1$ and $p_2$ are the sample proportions
• $n_1$ and $n_2$ are the sample sizes
• $Z_{\alpha/2}$ is the critical value from the standard normal distribution for a 90% confidence interval.

Step 2: Compute the Critical Value for 90% Confidence Interval

For a 90% confidence interval, $\alpha = 1 - 0.90 = 0.10$, and $\alpha/2 = 0.05$. The corresponding $Z$-score for a 90% confidence interval is:

$Z_{\alpha/2} = 1.645$

Step 3: Apply the Formula

Now, you'll need the values of $p_1$, $p_2$, $n_1$, and $n_2$ to proceed with the calculation. If these values are provided, we can plug them into the formula to calculate the confidence interval.

Step 4: Calculate the Confidence Interval

Using the formula:

$\text{Confidence Interval} = (p_1 - p_2) \pm 1.645 \cdot \sqrt{\frac{p_1(1 - p_1)}{n_1} + \frac{p_2(1 - p_2)}{n_2}}$

Finally, round the lower and upper bounds of the confidence interval to three decimal places.

Please provide the values of $p_1$, $p_2$, $n_1$, and $n_2$ to finalize the calculation. Would you like to provide these values or need help finding them?

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Related Questions:

1. How is the critical value $Z_{\alpha/2}$ determined for various confidence levels?
2. What factors influence the width of a confidence interval?
3. How do sample sizes affect the accuracy of a confidence interval?
4. Can a confidence interval be used to test a hypothesis? If so, how?
5. How does a confidence interval change if the confidence level is increased from 90% to 95%?

Tip: Always double-check your sample size and proportions before computing the confidence interval to avoid errors.