Drive-thru Time (See Problem 21 in Section 10.3.) The mean waiting time at the drive-thru of a fast-food restaurant from the time an order is placed to the time the order is received is 84.3 seconds. A manager devises a new drive-thru system that he believes will decrease wait time. He initiates the new system at his restaurant and measures the wait time for 10 randomly selected orders. The wait times are provided below.

a. The goal of the research is to determine if wait times have decreased as a result of the new system. Based on this research objective, state the null and alternative hypotheses.

Verified step by step guidance

1. Step 1: Identify the population mean \( \mu_0 \) from the problem. Here, the mean waiting time before the new system is \( \mu_0 = 84.3 \) seconds.
2. Step 2: Define the null hypothesis \( H_0 \) and the alternative hypothesis \( H_a \) based on the research goal. Since the manager wants to test if the new system decreases wait time, the hypotheses are:
3. \[ H_0: \mu = 84.3 \]
4. \[ H_a: \mu < 84.3 \]
5. where \( \mu \) is the mean waiting time after implementing the new system.
6. Step 3: Understand that this is a one-tailed test (left-tailed) because the alternative hypothesis is testing for a decrease (less than) in mean wait time.
7. Step 4: Note that the sample size is \( n = 10 \), and the sample data are the 10 wait times provided. These will be used to calculate the sample mean and sample standard deviation for further testing.
8. Step 5: Prepare to perform a hypothesis test (likely a t-test due to small sample size) comparing the sample mean to the population mean \( 84.3 \) seconds to determine if there is sufficient evidence to reject \( H_0 \).

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population parameter based on sample data. It involves stating a null hypothesis (no effect or difference) and an alternative hypothesis (effect or difference exists), then using sample data to determine whether to reject the null hypothesis.

One-Sample t-Test

A one-sample t-test compares the mean of a single sample to a known population mean when the population standard deviation is unknown. It helps determine if the sample mean significantly differs from the population mean, which is essential when testing if the new system reduces wait times.

Significance Level and Directional Hypotheses

The significance level (alpha) defines the threshold for rejecting the null hypothesis, commonly set at 0.05. Directional hypotheses specify the expected direction of the effect, such as a decrease in wait time, leading to a one-tailed test focused on detecting reductions rather than any change.