Step 1 :Define the null hypothesis as the proportion of females aged 15 and older living alone is still 55%, and the alternative hypothesis as the proportion has changed.
Step 2 :Calculate the sample proportion (p_hat) as the number of females living alone divided by the total sample size, which is \( \frac{242}{450} = 0.538 \).
Step 3 :Define the hypothesized proportion (p_0) as 0.55 and the sample size (n) as 450.
Step 4 :Calculate the test statistic (z) using the formula \( z = \frac{p_{hat} - p_{0}}{\sqrt{\frac{p_{0} \cdot (1 - p_{0})}{n}}} \), which gives \( z = -0.52 \).
Step 5 :Calculate the p-value based on the test statistic and the standard normal distribution, which gives a p-value of 0.60.
Step 6 :Since the p-value is greater than the significance level of 0.01, we do not reject the null hypothesis.
Step 7 :Therefore, there is not sufficient evidence at the 0.01 level of significance to conclude the proportion has changed.
Step 8 :Final Answer: \(\boxed{No}\)