Questions: Construct a confidence interval of the population proportion at the given level of confidence. x=860, n=1200, 94% confidence The lower bound of the confidence interval is . (Round to three decimal places as needed.) The upper bound of the confidence interval is . (Round to three decimal places as needed.)

Construct a confidence interval of the population proportion at the given level of confidence.

x=860, n=1200, 94% confidence

The lower bound of the confidence interval is .
(Round to three decimal places as needed.)
The upper bound of the confidence interval is .
(Round to three decimal places as needed.)

Transcript text: Construct a confidence interval of the population proportion at the given level of confidence. \[ x=860, n=1200,94 \% \text { confidence } \] The lower bound of the confidence interval is $\square$ . (Round to three decimal places as needed.) The upper bound of the confidence interval is $\square$ $\square$. (Round to three decimal places as needed.)

Solution

Solution Steps

Step 1: Calculate the Sample Proportion

The sample proportion $ \hat{p} $ is calculated using the formula: \[ \hat{p} = \frac{x}{n} \] where $ x = 860 $ (number of successes) and $ n = 1200 $ (sample size). Thus, \[ \hat{p} = \frac{860}{1200} = 0.717 \]

Step 2: Determine the Significance Level

The significance level $ \alpha $ is calculated as: \[ \alpha = 1 - \text{confidence level} = 1 - 0.94 = 0.06 \]

Step 3: Compute the Confidence Interval

Using the sample proportion $ \hat{p} $ and the significance level $ \alpha $, the confidence interval for the population proportion is computed. The resulting confidence interval is: \[ \left(0.692, 0.741\right) \]

Step 4: Identify the Bounds of the Confidence Interval

The lower bound of the confidence interval is: \[ \text{Lower Bound} = 0.692 \] The upper bound of the confidence interval is: \[ \text{Upper Bound} = 0.741 \]