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Construct a confidence interval for [tex]$P_1 - P_2$[/tex] at the given level of confidence.

[tex]x_1 = 25, \, n_1 = 267, \, x_2 = 33, \, n_2 = 282, \, 95\% \text{ confidence}[/tex]

The researchers are [tex]\square\%[/tex] confident the difference between the two population proportions, [tex]$p_1 - p_2$[/tex], is between [tex]\square[/tex] and [tex]\square[/tex].

(Use ascending order. Type an integer or decimal rounded to three decimal places as needed.)



Answer :

GinnyAnswer
Given the data:

- [tex]\( x_1 = 25 \)[/tex]
- [tex]\( n_1 = 267 \)[/tex]
- [tex]\( x_2 = 33 \)[/tex]
- [tex]\( n_2 = 282 \)[/tex]
- Confidence Level = 95%

We will construct a confidence interval for the difference between the two population proportions, [tex]\(p_1 - p_2\)[/tex].

### Step-by-Step Solution:

1. Calculate Sample Proportions:

[tex]\[ \hat{p}_1 = \frac{x_1}{n_1} = \frac{25}{267} \approx 0.094 \][/tex]

[tex]\[ \hat{p}_2 = \frac{x_2}{n_2} = \frac{33}{282} \approx 0.117 \][/tex]

2. Calculate the Combined Standard Error:

[tex]\[ \text{Standard Error} = \sqrt{\left(\frac{\hat{p}_1 (1 - \hat{p}_1)}{n_1}\right) + \left(\frac{\hat{p}_2 (1 - \hat{p}_2)}{n_2}\right)} \][/tex]

[tex]\[ \text{Standard Error} \approx 0.026 \][/tex]

3. Determine the Critical Value for a 95% Confidence Level:

The critical value for [tex]\( 95\% \)[/tex] confidence using the standard normal distribution is approximately [tex]\( 1.960 \)[/tex].

4. Calculate the Margin of Error:

[tex]\[ \text{Margin of Error} = \text{Critical Value} \times \text{Standard Error} \][/tex]

[tex]\[ \text{Margin of Error} \approx 1.960 \times 0.026 = 0.051 \][/tex]

5. Calculate the Confidence Interval:

Difference in Sample Proportions:

[tex]\[ \hat{p}_1 - \hat{p}_2 = 0.094 - 0.117 \approx -0.023 \][/tex]

Calculate the lower and upper bounds:

[tex]\[ \text{Lower Bound} = (\hat{p}_1 - \hat{p}_2) - \text{Margin of Error} \approx -0.023 - 0.051 \approx -0.075 \][/tex]

[tex]\[ \text{Upper Bound} = (\hat{p}_1 - \hat{p}_2) + \text{Margin of Error} \approx -0.023 + 0.051 \approx 0.028 \][/tex]

### Conclusion:

The researchers are [tex]\( 95 \% \)[/tex] confident the difference between the two population proportions, [tex]\(p_1 - p_2\)[/tex], is between [tex]\(-0.075\)[/tex] and [tex]\(0.028\)[/tex].

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