To construct a confidence interval for the true population proportion, we can use the following formula for a confidence interval of a population proportion:

$\hat{p} \pm Z_{\alpha/2} \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$

Where:

• $\hat{p}$ is the sample proportion.
• $Z_{\alpha/2}$ is the critical value from the standard normal distribution for the desired confidence level.
• $n$ is the sample size.

Step 1: Calculate the sample proportion $\hat{p}$

Given that 270 out of 300 people have kids: $\hat{p} = \frac{270}{300} = 0.9$

Step 2: Find the $Z_{\alpha/2}$ value for a 90% confidence level

For a 90% confidence interval, the $\alpha$ value is 0.10 (because $100\% - 90\% = 10\%$, and 5% is in each tail). The critical value $Z_{\alpha/2}$ corresponds to the z-value for 95% of the data under the normal curve, which is:

$Z_{\alpha/2} = 1.645$

Step 3: Calculate the standard error

The standard error (SE) of the sample proportion is:

$SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} = \sqrt{\frac{0.9(1 - 0.9)}{300}} = \sqrt{\frac{0.9 \times 0.1}{300}} = \sqrt{\frac{0.09}{300}} = \sqrt{0.0003} \approx 0.01732$

Step 4: Construct the confidence interval

Now, the confidence interval is:

$\hat{p} \pm Z_{\alpha/2} \times SE$

Substitute the values:

$0.9 \pm 1.645 \times 0.01732$

$0.9 \pm 0.02846$

Thus, the 90% confidence interval is:

$0.9 - 0.02846 = 0.8715$ $0.9 + 0.02846 = 0.9285$

Final Answer:

The 90% confidence interval for the true population proportion is:

$0.872 < p < 0.929$

Let me know if you'd like further clarification or have any questions!

Follow-up Questions:

1. How would you calculate the confidence interval for a different confidence level (e.g., 95%)?
2. What assumptions do we need to make when using this formula for proportions?
3. How would the confidence interval change if the sample size were larger (e.g., 500 people)?
4. Why is the z-value used instead of the t-value for constructing confidence intervals for proportions?
5. What effect would a smaller sample proportion (e.g., 0.6 instead of 0.9) have on the confidence interval width?

Tip:

As sample size increases, the confidence interval becomes narrower, meaning we can estimate the population proportion more accurately.