 Test the claim by constructing an appropriate confidence interval.
The 90​% confidence interval is 
  
enter your response hereless thanleft parenthesis p 1 minus p 2 right parenthesisless than
  
enter your response here.
​(Round to three decimal places as​ needed.) 

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.

Test the claim by constructing an appropriate confidence interval. The 90% confidence interval is less than (p1 - p2) less than (Round to three decimal places as needed.)

Learn how to construct a 90% confidence interval for the difference in proportions using the formula (p1 - p2) ± Z * sqrt[(p1(1 - p1)/n1) + (p2(1 - p2)/n2)]. This problem explores key statistical concepts including proportions and Z-scores. Suitable for college-level statistics students.

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