Two samples are taken with the following numbers of successes and sample sizes x1 = 36 x2 = 23 n1 = 59 n2 = 53 Find a 96% confidence interval, round answers to the nearest thousandth. < pi – P2 <

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### Confidence Interval Calculation for Proportion Differences

#### Given Data:

Two samples are taken with the following numbers of successes and sample sizes:

- \( x_1 = 36 \)
- \( x_2 = 23 \)
- \( n_1 = 59 \)
- \( n_2 = 53 \)

#### Objective:

Find a 96% confidence interval for the difference in proportions \( p_1 - p_2 \). Round the answers to the nearest thousandth.

#### Instructions:

- Calculate the sample proportions: \( \hat{p}_1 \) and \( \hat{p}_2 \).
- Use the formula for the confidence interval of the difference between two proportions.
  
#### Formula:

\[
\text{Confidence Interval} = (\hat{p}_1 - \hat{p}_2) \pm Z \sqrt{\left(\frac{\hat{p}_1(1 - \hat{p}_1)}{n_1}\right) + \left(\frac{\hat{p}_2(1 - \hat{p}_2)}{n_2}\right)}
\]

Where:
- \( \hat{p}_1 = \frac{x_1}{n_1} \)
- \( \hat{p}_2 = \frac{x_2}{n_2} \)
- \( Z \) is the Z-value corresponding to the desired confidence level (for 96%, Z ≈ 2.05).

#### Steps:

1. Calculate \( \hat{p}_1 \) and \( \hat{p}_2 \):
   \[
   \hat{p}_1 = \frac{36}{59}, \quad \hat{p}_2 = \frac{23}{53}
   \]

2. Find the standard error (SE) for the difference in proportions:
   \[
   SE = \sqrt{\left(\frac{\hat{p}_1(1 - \hat{p}_1)}{59}\right) + \left(\frac{\hat{p}_2(1 - \hat{p}_2)}{53}\right)}
   \]

3. Multiply the SE by the Z-value for 96% confidence and apply to the difference in sample proportions to find the confidence interval:
   \[
   \text{Confidence Interval} = (\hat{p}_1 - \hat{p}_2) \pm
Transcribed Image Text:### Confidence Interval Calculation for Proportion Differences #### Given Data: Two samples are taken with the following numbers of successes and sample sizes: - \( x_1 = 36 \) - \( x_2 = 23 \) - \( n_1 = 59 \) - \( n_2 = 53 \) #### Objective: Find a 96% confidence interval for the difference in proportions \( p_1 - p_2 \). Round the answers to the nearest thousandth. #### Instructions: - Calculate the sample proportions: \( \hat{p}_1 \) and \( \hat{p}_2 \). - Use the formula for the confidence interval of the difference between two proportions. #### Formula: \[ \text{Confidence Interval} = (\hat{p}_1 - \hat{p}_2) \pm Z \sqrt{\left(\frac{\hat{p}_1(1 - \hat{p}_1)}{n_1}\right) + \left(\frac{\hat{p}_2(1 - \hat{p}_2)}{n_2}\right)} \] Where: - \( \hat{p}_1 = \frac{x_1}{n_1} \) - \( \hat{p}_2 = \frac{x_2}{n_2} \) - \( Z \) is the Z-value corresponding to the desired confidence level (for 96%, Z ≈ 2.05). #### Steps: 1. Calculate \( \hat{p}_1 \) and \( \hat{p}_2 \): \[ \hat{p}_1 = \frac{36}{59}, \quad \hat{p}_2 = \frac{23}{53} \] 2. Find the standard error (SE) for the difference in proportions: \[ SE = \sqrt{\left(\frac{\hat{p}_1(1 - \hat{p}_1)}{59}\right) + \left(\frac{\hat{p}_2(1 - \hat{p}_2)}{53}\right)} \] 3. Multiply the SE by the Z-value for 96% confidence and apply to the difference in sample proportions to find the confidence interval: \[ \text{Confidence Interval} = (\hat{p}_1 - \hat{p}_2) \pm
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