Construct a 95% confidence interval of the population proportion using the given information. x = 105, n = 150 Click here to view the table of critical values. The lower bound is. Table of critical values The upper bound is. (Round to three decimal places as needed.) a Level of Confidence, (1 - a) · 100% Area in Each Tail, Critical Value, z; 90% 0,05 1.645 95% 0,025 1.96 99% 0,005 2.575 Print Done

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To construct a 95% confidence interval for the population proportion using the given information, please follow these instructions:

Given:
- x = 105
- n = 150

### Steps:
1. **Calculate the Sample Proportion (p̂):**
   \[
   p̂ = \frac{x}{n} = \frac{105}{150}
   \]

2. **Find the Critical Value (zₐ/₂):**
   - Refer to the table of critical values for a 95% confidence level:
     - **Level of Confidence (1 - α) %:** 95%
     - **Area in Each Tail (α/2):** 0.025
     - **Critical Value (zₐ/₂):** 1.96

3. **Compute the Standard Error (SE) of the Proportion:**
   \[
   SE = \sqrt{\frac{p̂(1 - p̂)}{n}}
   \]

4. **Calculate the Confidence Interval:**
   - **Lower Bound:** \( p̂ - zₐ/₂ \times SE \)
   - **Upper Bound:** \( p̂ + zₐ/₂ \times SE \)

### Note:
- Round the final answers to three decimal places as needed.

### Diagram Explanation:
- **Table of Critical Values:** This table provides critical values for various levels of confidence. For a 95% confidence interval, the critical value is 1.96, with an area of 0.025 in each tail of the normal distribution.

Remember to substitute the calculated values into the formulas to get the specific lower and upper bounds of your confidence interval.
Transcribed Image Text:To construct a 95% confidence interval for the population proportion using the given information, please follow these instructions: Given: - x = 105 - n = 150 ### Steps: 1. **Calculate the Sample Proportion (p̂):** \[ p̂ = \frac{x}{n} = \frac{105}{150} \] 2. **Find the Critical Value (zₐ/₂):** - Refer to the table of critical values for a 95% confidence level: - **Level of Confidence (1 - α) %:** 95% - **Area in Each Tail (α/2):** 0.025 - **Critical Value (zₐ/₂):** 1.96 3. **Compute the Standard Error (SE) of the Proportion:** \[ SE = \sqrt{\frac{p̂(1 - p̂)}{n}} \] 4. **Calculate the Confidence Interval:** - **Lower Bound:** \( p̂ - zₐ/₂ \times SE \) - **Upper Bound:** \( p̂ + zₐ/₂ \times SE \) ### Note: - Round the final answers to three decimal places as needed. ### Diagram Explanation: - **Table of Critical Values:** This table provides critical values for various levels of confidence. For a 95% confidence interval, the critical value is 1.96, with an area of 0.025 in each tail of the normal distribution. Remember to substitute the calculated values into the formulas to get the specific lower and upper bounds of your confidence interval.
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