Question

In a survey consisting of a simple random sample of n = 1,217 households, 837 households included at least one child. (a) Construct a 90% confidence interval to estimate the proportion of all.

Answer 1

To construct the confidence interval for the proportion of households with at least one child, use the formula: Confidence Interval = Sample Proportion ± (Z * Standard Error). Calculate the sample proportion and standard error using the given information, and substitute into the formula to find the confidence interval. The 90% confidence interval to estimate the proportion is (0.6646, 0.7096).

Step-by-step explanation:

To construct the confidence interval to estimate the proportion of all households with at least one child, we can use the formula:

Confidence Interval = Sample Proportion ± (Z * Standard Error)

Given that the sample size is 1,217 and 837 households included at least one child, the sample proportion is 837/1217 = 0.6871. The Z-value for a 90% confidence level is 1.645.

The standard error can be calculated using the formula: Standard Error = sqrt((p*(1-p))/n), where p is the sample proportion and n is the sample size.

Substituting the values, the standard error is approximately 0.0143.

Therefore, the confidence interval is given by: 0.687 ± (1.645 * 0.0143), which simplifies to (0.6646, 0.7096).