Out of 400 people sampled, 308 had kids. Assuming the conditions are met, construct a theoretical 99% confidence interval for the true population proportion of people with kids.

Give your answers as decimals, to four places.

(  ,  )

To construct a 99% confidence interval for the true population proportion of people with kids, we follow these steps:

### 1. Identify the sample proportion (\( \hat{p} \)):
The sample proportion (\( \hat{p} \)) is calculated as:
\[
\hat{p} = \frac{\text{number of people with kids}}{\text{total sample size}} = \frac{308}{400} = 0.77
\]

### 2. Determine the critical value for a 99% confidence interval:
For a 99% confidence interval, the critical value (\( z_{\alpha/2} \)) corresponds to the Z-value where the area under the standard normal curve between -\( z_{\alpha/2} \) and \( z_{\alpha/2} \) is 99%. This value is approximately:
\[
z_{\alpha/2} = 2.576
\]

### 3. Compute the standard error (SE):
The standard error (SE) of the sample proportion is calculated using the formula:
\[
SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} = \sqrt{\frac{0.77(1 - 0.77)}{400}} = \sqrt{\frac{0.77 \times 0.23}{400}} = \sqrt{\frac{0.1771}{400}} \approx 0.0210
\]

### 4. Calculate the margin of error (ME):
The margin of error is given by:
\[
ME = z_{\alpha/2} \times SE = 2.576 \times 0.0210 \approx 0.0541
\]

### 5. Construct the confidence interval:
The confidence interval is:
\[
\hat{p} \pm ME = 0.77 \pm 0.0541
\]
Thus, the confidence interval is:
\[
(0.7700 - 0.0541, 0.7700 + 0.0541) = (0.7159, 0.8241)
\]

### Final Answer:
The 99% confidence interval for the true population proportion is:
\[
(0.7159, 0.8241)
\]

Would you like further details or have any questions?

Here are five follow-up questions you could explore:
1. How would the confidence interval change if the sample size increased?
2. What is the interpretation of this confidence interval in context?
3. How would a 95% confidence interval differ from a 99% one in this case?
4. How does the sample proportion (\( \hat{p} \)) affect the width of the interval?
5. How does the standard error influence the margin of error?

**Tip:** A larger sample size reduces the margin of error, leading to a narrower confidence interval, all else being equal.

Out of 400 people sampled, 308 had kids. Assuming the conditions are met, construct a theoretical 99% confidence interval for the true population proportion of people with kids. Give your answers as decimals, to four places. (  ,  )

Learn how to calculate the 99% confidence interval for the true population proportion of people with kids based on a sample of 400 people, 308 of whom have children. This problem covers key statistical concepts such as sample proportions, standard errors, and confidence intervals. Suitable for college-level statistics or AP Statistics students.

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