You are the manager of a restaurant that delivers pizza to customers. You have just changed your delivery process in an effort to reduce the mean time between the order and completion of delivery from the current 26 minutes. From past​ experience, you can assume that the population standard deviation is 7 minutes. A sample of 49 orders using the new delivery process yields a sample mean of 23.29 minutes. The level of significance is 0.01 and the sample size is 49. Use the​ Z-test statistic, and use the normal distribution for a sampling distribution. Determine the critical​ value(s) that​ divide(s) the rejection and​ non-rejection region(s). The critical​ value(s) is(are) 

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!

---

### 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.

You are the manager of a restaurant that delivers pizza to customers. You have just changed your delivery process in an effort to reduce the mean time between the order and completion of delivery from the current 26 minutes. From past​ experience, you can assume that the population standard deviation is 7 minutes. A sample of 49 orders using the new delivery process yields a sample mean of 23.29 minutes. The level of significance is 0.01 and the sample size is 49. Use the​ Z-test statistic, and use the normal distribution for a sampling distribution. Determine the critical​ value(s) that​ divide(s) the rejection and​ non-rejection region(s). The critical​ value(s) is(are).

This problem involves hypothesis testing to determine if a new pizza delivery process significantly reduces the mean delivery time from 26 minutes to 23.29 minutes, with a sample size of 49 and a population standard deviation of 7 minutes. Using a Z-test with a significance level of 0.01, learn how to calculate the critical value and interpret the results.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation