Question
A sample of voters in a certain city was asked whether they planned to vote for the incumbent mayor in the next election. The following display from a TI-84 Plus calculator presents a 99% confidence interval for the population proportion that plan to vote for the incumbent mayor.
1-PropZInt
(.34795, .66743)
^p=.5076923077
n = 65
a) Fill in the blanks. We are ______% confident that the population proportion is between ______ and ______. Round the answers to three decimal places as needed.
1-PropZInt
(.34795, .66743)
^p=.5076923077
n = 65
a) Fill in the blanks. We are ______% confident that the population proportion is between ______ and ______. Round the answers to three decimal places as needed.
Answers
Answer
(b) Use the information in the display to construct a 95% confidence interval for p. Round the answers to three decimal places.
Answered by
GPT-4o mini
To fill in the blanks based on the provided confidence interval display, we can extract the necessary information from the interval and context:
- The confidence level is given as 99%, so we are **99%** confident.
- The lower bound of the confidence interval is **0.34795**, which rounds to **0.348** to three decimal places.
- The upper bound of the confidence interval is **0.66743**, which rounds to **0.667** to three decimal places.
Therefore, the filled-in statement will read:
We are **99%** confident that the population proportion is between **0.348** and **0.667**.
- The confidence level is given as 99%, so we are **99%** confident.
- The lower bound of the confidence interval is **0.34795**, which rounds to **0.348** to three decimal places.
- The upper bound of the confidence interval is **0.66743**, which rounds to **0.667** to three decimal places.
Therefore, the filled-in statement will read:
We are **99%** confident that the population proportion is between **0.348** and **0.667**.
Answered by
GPT-4o mini
To construct a 95% confidence interval for the population proportion \( p \), we will use the sample proportion and the standard error of the sample proportion. We know the following from the given information:
- Sample proportion \( \hat{p} = 0.5077 \) (as provided in the display).
- Sample size \( n = 65 \).
Now, to calculate the 95% confidence interval, we will take the following steps:
1. **Calculate the standard error (SE)** of the sample proportion:
\[
SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} = \sqrt{\frac{0.5076923077 \times (1 - 0.5076923077)}{65}}.
\]
2. **Calculate the critical value (z*) for 95% confidence**:
The critical z-value for a 95% confidence level is approximately 1.96.
3. **Use the formula for the confidence interval**:
\[
CI = \hat{p} \pm z^* \cdot SE.
\]
Now, let's go through the calculations:
1. **Calculate SE**:
\[
SE = \sqrt{\frac{0.5077 \times (1 - 0.5077)}{65}} \approx \sqrt{\frac{0.5077 \times 0.4923}{65}} \approx \sqrt{\frac{0.249173911}{65}} \approx \sqrt{0.003833} \approx 0.0619.
\]
2. **Calculate the confidence interval**:
\[
CI = 0.5077 \pm 1.96 \cdot 0.0619.
\]
Calculate the margin of error:
\[
Margin of Error = 1.96 \times 0.0619 \approx 0.1211.
\]
Now calculate the lower and upper bounds of the interval:
\[
Lower Bound = 0.5077 - 0.1211 \approx 0.3866,
\]
\[
Upper Bound = 0.5077 + 0.1211 \approx 0.6288.
\]
3. **Round to three decimal places**:
- Lower Bound: 0.387
- Upper Bound: 0.629
Thus, the 95% confidence interval for \( p \) is:
**(0.387, 0.629)**.
- Sample proportion \( \hat{p} = 0.5077 \) (as provided in the display).
- Sample size \( n = 65 \).
Now, to calculate the 95% confidence interval, we will take the following steps:
1. **Calculate the standard error (SE)** of the sample proportion:
\[
SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} = \sqrt{\frac{0.5076923077 \times (1 - 0.5076923077)}{65}}.
\]
2. **Calculate the critical value (z*) for 95% confidence**:
The critical z-value for a 95% confidence level is approximately 1.96.
3. **Use the formula for the confidence interval**:
\[
CI = \hat{p} \pm z^* \cdot SE.
\]
Now, let's go through the calculations:
1. **Calculate SE**:
\[
SE = \sqrt{\frac{0.5077 \times (1 - 0.5077)}{65}} \approx \sqrt{\frac{0.5077 \times 0.4923}{65}} \approx \sqrt{\frac{0.249173911}{65}} \approx \sqrt{0.003833} \approx 0.0619.
\]
2. **Calculate the confidence interval**:
\[
CI = 0.5077 \pm 1.96 \cdot 0.0619.
\]
Calculate the margin of error:
\[
Margin of Error = 1.96 \times 0.0619 \approx 0.1211.
\]
Now calculate the lower and upper bounds of the interval:
\[
Lower Bound = 0.5077 - 0.1211 \approx 0.3866,
\]
\[
Upper Bound = 0.5077 + 0.1211 \approx 0.6288.
\]
3. **Round to three decimal places**:
- Lower Bound: 0.387
- Upper Bound: 0.629
Thus, the 95% confidence interval for \( p \) is:
**(0.387, 0.629)**.
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