In 2011, a U.S. Census report determined that \( 55 \% \) of college students are working students. A researcher thinks this percentage has changed and surveys 176 college students. The researcher reports that 122 of the 176 are working students. Is there evidence to support the researcher's claim at the \( 1 \% \) significance level? a. Determine the null and alternative hypotheses. \( H_{0}: p= \) (Select the correct symbol and enter the value.) \( H_{1}: p \) ?v Determine the test statistic. Round to two decimal places. \( z= \) c. Find the \( p \) - value. Round to four decimal places. b - value = d. Are the results statistically significant? Oail to reject the null hypothesis. Reject the null hypothesis. e. Write the conclusion. There is not sufficient evidence to support the claim that the percentage of working college students has changed. There is sufficient evidence to support the claim that the percentage of working college students has changed.

The null hypothesis states that the proportion of working students is equal to the original percentage established in 2011. Hence, we have \( H_{0}: p = 0.55 \). The alternative hypothesis suggests that the proportion has changed, which leads us to \( H_{1}: p \neq 0.55 \). Now onto the calculations: the sample proportion (\( \hat{p} \)) is \( \frac{122}{176} \approx 0.6932 \). The test statistic \( z \) can be calculated using the formula: \[ z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}} \] Substituting the values, we find \( z \) approximately equals 4.60 (rounded to two decimal places). For the \( p \)-value, this two-tailed test gives us \( p \)-value ≈ 0.0000 (rounded to four decimal places). Since the \( p \)-value (0.0000) is less than the significance level (0.01), we reject the null hypothesis. In conclusion, there is sufficient evidence to support the claim that the percentage of working college students has changed. Now for some fun tidbits: Did you know that historically, the rate of working students has been fluctuating depending on economic conditions? This is often due to financial needs, tuition increases, and overall employment trends. It ties closely to larger societal changes, making the educational landscape a fascinating field of study! If you're interested in diving deeper into this topic, check out "The College Student’s Guide to Job Opportunities" which explores how students balance work and study effectively. It discusses not just statistics but personal stories that resonate with the challenges many students face today. Happy reading!