In this problem, we're conducting a hypothesis test to determine whether the new delivery process has significantly reduced the mean delivery time from the previous mean of 26 minutes.

Step 1: Set up the hypotheses

• Null Hypothesis $(H_0)$: The mean delivery time has not decreased, i.e., $\mu = 26$ minutes.
• Alternative Hypothesis $(H_a)$: The mean delivery time has decreased, i.e., $\mu < 26$ minutes.

Since the question indicates that we are looking for a decrease, this is a one-tailed test (specifically, a left-tailed test).

Step 2: Identify the significance level

The level of significance is $\alpha = 0.01$, meaning that we are conducting the test with a 1% risk of rejecting the null hypothesis when it is true.

Step 3: Calculate the critical value(s)

To find the critical value, we need to determine the $Z$-value that corresponds to the $\alpha = 0.01$ level of significance in the left tail of the standard normal distribution.

Since this is a one-tailed test (left-tailed), the critical value will be the $Z$-score that leaves 1% of the area in the left tail of the normal distribution.

Using a standard normal distribution table or a calculator, the critical value for $\alpha = 0.01$ in a left-tailed test is approximately:

$Z_{\alpha} = -2.33$

Step 4: Conclusion

The critical value that divides the rejection and non-rejection regions is approximately -2.33. If the test statistic $Z$ is less than -2.33, we will reject the null hypothesis $H_0$; otherwise, we fail to reject it.

Let me know if you'd like to proceed with the next steps, such as calculating the test statistic and drawing a conclusion!

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Follow-up questions:

1. How do we calculate the test statistic in a hypothesis test?
2. What is the importance of the level of significance ($\alpha$) in hypothesis testing?
3. How do we interpret a one-tailed versus a two-tailed test?
4. What role does the population standard deviation play in the $Z$-test?
5. How would the result change if the significance level was set at $\alpha = 0.05$?

Tip: Always sketch a normal distribution curve when conducting $Z$-tests to visually understand the critical region.