If n=130 and 𝑝^ (p-hat) = 0.08, construct a 95% confidence interval.Give your answers to two decimals < p <

1. Break Down the Problem

We need to construct a 95% confidence interval for the population proportion $p$ based on the sample proportion $\hat{p}$ and the sample size $n$.

2. Relevant Concepts

To construct the confidence interval for the population proportion, we use the formula:

$\hat{p} \pm z \cdot \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$

Where:

• $\hat{p} = 0.08$ (sample proportion)
• $n = 130$ (sample size)
• $z$ is the z-score corresponding to the desired confidence level (for 95%, $z \approx 1.96$)

3. Analysis and Detail

1. Calculate the standard error (SE): $SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} = \sqrt{\frac{0.08(1 - 0.08)}{130}}$ $SE = \sqrt{\frac{0.08 \cdot 0.92}{130}} = \sqrt{\frac{0.0736}{130}} = \sqrt{0.0005661538} \approx 0.0238$

2. Calculate the margin of error (ME): $ME = z \cdot SE = 1.96 \cdot 0.0238 \approx 0.0467$

3. Construct the confidence interval: $\text{Lower limit} = \hat{p} - ME = 0.08 - 0.0467 \approx 0.0333$ $\text{Upper limit} = \hat{p} + ME = 0.08 + 0.0467 \approx 0.1267$

4. Verify and Summarize

The final step is summarizing the results:

• Lower limit: $0.0333$
• Upper limit: $0.1267$

Thus, the confidence interval can be rounded to two decimal places.

Final Answer

The 95% confidence interval is:

$0.03 < p < 0.13$