Bootstrapping methods were used to construct a \( 95 \% \) confidence interval for the mean monthly rent for a one-bedroom apartment in Manhattan of \( \$ 2,649.68, \$ 3,814.63 \) ]. Later, someone looking for a one-bedroom apartment asks if the average monthly rent is different from \( \$ 2,500 \). The following hypotheses were drafted: \( H_{0}: \mu=\$ 2500 \) \( H_{a}: \mu \neq \$ 2500 \) If the hypotheses above were tested at the 0.05 alpha level using the same dataset, would we expect to reject or fail to reject the null hypothesis? O Reject the null hypothesis O Fall to reject the null hypothesis

Given that the \( 95\% \) confidence interval for the mean monthly rent is \( [\$ 2,649.68, \$ 3,814.63] \), it does not include \( \$ 2,500 \). Hence, if you were to conduct a hypothesis test at the \( 0.05 \) alpha level, you'd likely reject the null hypothesis \( H_{0}: \mu = \$ 2,500 \) because the sample mean is significantly different from \( \$ 2,500 \). Now, just remember that a \( 95\% \) confidence interval implies if you took multiple samples, \( 95\% \) of the intervals calculated from those samples would contain the true population mean. Since \( \$ 2,500 \) falls outside the constructed confidence interval, you have solid evidence that the average rent is likely different from that figure. So gear up for a rejection party for that null hypothesis! 🎉