Sure, let's go through the process step by step to construct a 95% confidence interval for the true population proportion of adults with children.
### Step-by-Step Solution:
1. Sample Size ([tex]\( n \)[/tex]):
- The sample size [tex]\( n \)[/tex] is given as 490.
2. Number of Adults with Children:
- Out of these 490 adults, 343 have children.
3. Sample Proportion ([tex]\( \hat{p} \)[/tex]):
- The sample proportion [tex]\( \hat{p} \)[/tex] is calculated as the number of adults with children divided by the total sample size:
[tex]\[
\hat{p} = \frac{343}{490} = 0.7
\][/tex]
4. Z-Value for 95% Confidence:
- For a 95% confidence interval, we use a Z-value corresponding to the middle 95% of the standard normal distribution. The Z-value for 95% confidence is approximately 1.96.
5. Standard Error of the Proportion ([tex]\( SE \)[/tex]):
- The standard error of the proportion is calculated using the formula:
[tex]\[
SE = \sqrt{\frac{\hat{p} (1 - \hat{p})}{n}}
\][/tex]
Plugging in the values, we get:
[tex]\[
SE = \sqrt{\frac{0.7 \times (1 - 0.7)}{490}} = 0.0207
\][/tex]
6. Margin of Error (ME):
- The margin of error is calculated by multiplying the Z-value by the standard error:
[tex]\[
ME = Z \times SE = 1.96 \times 0.0207 = 0.0406
\][/tex]
7. Confidence Interval:
- The confidence interval is calculated by adding and subtracting the margin of error from the sample proportion:
[tex]\[
\text{Lower Limit} = \hat{p} - ME = 0.7 - 0.0406 = 0.659
\][/tex]
[tex]\[
\text{Upper Limit} = \hat{p} + ME = 0.7 + 0.0406 = 0.741
\][/tex]
Therefore, the 95% confidence interval for the true population proportion of adults with children is approximately (0.659, 0.741).
This means that we are 95% confident that the true proportion of adults with children in the population is between 0.659 and 0.741.
