Questions: Working at home: According to the U.S. Census Bureau, 42% of men who worked at home were college graduates. In a sample of 480 women who worked at home, 157 were college graduates. Part 1 of 3 (a) Find a point estimate for the proportion of college graduates among women who work at home. Round the answer to at least three decimal places. Part 2 of 3 (b) Construct a 95% confidence interval for the proportion of women who work at home who are college graduates. Round the answer to at least three decimal places.

Solution Steps

To find the point estimate for the proportion of college graduates among women who work at home, we use the formula:

\[ \hat{p} = \frac{x}{n} \]

where:

• \( x = 157 \) (number of college graduates)
• \( n = 480 \) (total number of women in the sample)

Calculating the point estimate:

\[ \hat{p} = \frac{157}{480} \approx 0.327 \]

Thus, the point estimate for the proportion of college graduates among women who work at home is \( 0.327 \).

To construct a \( 95\% \) confidence interval for the proportion of women who work at home who are college graduates, we use the formula:

\[ \hat{p} \pm z \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \]

where:

• \( \hat{p} = 0.327 \)
• \( z \) is the z-score corresponding to a \( 95\% \) confidence level, which is \( 1.96 \)
• \( n = 480 \)

Calculating the margin of error:

\[ \text{Margin of Error} = 1.96 \cdot \sqrt{\frac{0.327(1 - 0.327)}{480}} \approx 0.042 \]

Now, we can calculate the confidence interval:

\[ \text{Lower Limit} = \hat{p} - \text{Margin of Error} \approx 0.327 - 0.042 \approx 0.285 \] \[ \text{Upper Limit} = \hat{p} + \text{Margin of Error} \approx 0.327 + 0.042 \approx 0.369 \]

Thus, the \( 95\% \) confidence interval for the proportion of women who work at home who are college graduates is \( (0.285, 0.369) \).

Final Answer