To determine the confidence level $C$, we need to calculate the $z$-score corresponding to the given confidence interval bounds. Here's the process:

Step 1: Identify the sample proportion and margin of error

The confidence interval is given as $0.565$ to $0.695$. From this:

• The sample proportion ($\hat{p}$) is the midpoint of the interval: $\hat{p} = \frac{0.565 + 0.695}{2} = 0.63$
• The margin of error ($ME$) is the distance from the sample proportion to either endpoint of the interval: $ME = 0.695 - 0.63 = 0.065$

Step 2: Express the margin of error formula

The margin of error for a proportion is: $ME = z^* \cdot \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$ Where:

• $z^*$ is the critical value associated with the confidence level $C$,
• $n = 100$ is the sample size,
• $\hat{p} = 0.63$ is the sample proportion.

Step 3: Solve for $z^*$

Substitute the known values into the margin of error formula: $0.065 = z^* \cdot \sqrt{\frac{0.63(1 - 0.63)}{100}}$ First, calculate the standard error: $\sqrt{\frac{0.63(1 - 0.63)}{100}} = \sqrt{\frac{0.63 \cdot 0.37}{100}} = \sqrt{0.2331 / 100} = \sqrt{0.002331} \approx 0.0483$ Then solve for $z^*$: $z^* = \frac{0.065}{0.0483} \approx 1.346$

Step 4: Find the confidence level

The $z^*$-value of approximately $1.346$ corresponds to a confidence level $C$. Using standard normal distribution tables or software, we find that the area between $-1.346$ and $1.346$ is approximately $0.824$, or 82.4%.

Thus, the confidence level $C$ is 82.4%.

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Would you like a detailed explanation of using the $z$-table, or do you have further questions? 😊

Expanding Your Knowledge:

1. How is the confidence interval affected by the sample size $n$?
2. What happens to the margin of error if the confidence level $C$ increases?
3. How do you interpret a confidence interval in real-world terms?
4. Can confidence intervals be used for means as well as proportions? If so, how?
5. How do we determine the sample size needed for a specific margin of error?

Tip: Confidence intervals are a way to estimate population parameters using sample data—be sure to always check assumptions like random sampling and sufficient sample size!