A transportation company is concerned that recent changes to road conditions might have increased the average commute time for its delivery drivers.

Drivers previously averaged 30 minutes for their commute.
Formulate the correct null and alternative hypotheses to test whether the average commute time has increased.

$\begin{array}{l}
H_0: \mu>30 \\
H_a: \mu=30
\end{array}$

$\begin{array}{l}
H_0: \mu<30 \\
H_a: \mu>

- The null hypothesis \(H_0\) represents the status quo, stating that the average commute time remains 30 minutes: \(H_0: \mu = 30\). - The alternative hypothesis \(H_a\) represents the company's concern that the average commute time has increased: \(H_a: \mu > 30\). - Therefore, the correct formulation is \(H_0: \mu = 30\) and \(H_a: \mu > 30\). - The final answer is \(\boxed{{H_0: \mu = 30 \text{ and } H_a: \mu > 30}}\). ### Explanation 1. Analyze the problem and identify the objective The transportation company is concerned that the average commute time for its delivery drivers might have increased due to recent changes in road conditions. Previously, the average commute time was 30 minutes. Our goal is to formulate the null and alternative hypotheses to statistically test if this average commute time has indeed increased. 2. Define the null hypothesis (H₀) The null hypothesis, denoted as \(H_0\), represents the status quo or the absence of an effect. In this case, the status quo is that the average commute time has not changed from its previous value of 30 minutes. Therefore, the null hypothesis states that the population mean commute time (\(\mu\)) is equal to 30 minutes. \[H_0: \mu = 30\] 3. Define the alternative hypothesis (Hₐ) The alternative hypothesis, denoted as \(H_a\) (or \(H_1\)), represents the claim or the effect that we are trying to find evidence for. The company is concerned that the average commute time has *increased*. Therefore, the alternative hypothesis states that the population mean commute time (\(\mu\)) is greater than 30 minutes. \[H_a: \mu > 30\] 4. Select the correct option Based on our formulation, the correct set of null and alternative hypotheses is: \[H_0: \mu = 30\] \[H_a: \mu > 30\] Comparing this with the given options, the third option matches our derived hypotheses. ### Examples Understanding how to formulate null and alternative hypotheses is crucial in many real-world scenarios. For instance, a pharmaceutical company might use this to test if a new drug significantly reduces blood pressure (\(H_a: \mu < \text{baseline}\)). Or, an educator might test if a new teaching method improves test scores (\(H_a: \mu > \text{previous average}\)). This statistical framework allows us to make data-driven decisions and draw meaningful conclusions from observations.