105.3 79.0 The mean waiting time at the drive-through of a fast-food restaurant from the time an order is placed to the time the order is received is 84.7 seconds. A manager devises a new drive-through system that he believes will decrease wait time. As a test, he initiates the new system at his restaurant and measures the wait time for 10 randomly selected orders. The wait times are provided in the table to the right. Complete parts (a) and (b) below. 66.4 56.7 95.7 84.7 75.0 66.5 71.1 79.6 E Click the icon to view the table of correlation coefficient critical values. (a) Because the sample size is small, the manager must verify that the wait time is normally distributed and the sample does not contain any outliers. The normal probability plot is shown below and the sample correlation coefficient is known to be r= 0.981. Are the conditions for testing the hypothesis satisfied? Yes, the conditions satisfied. The normal probability plot is linear enough, since the correlation coefficient is AExpected z-score Q are greater than the critical value. In addition, a boxplot does not show any outliers. 1- 0- 60 5 90 165 -1- Time (sec) (b) Is the new system effective? Conduct a hypothesis test using the P-value approach and a level of significance of a = 0.01.

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

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The mean waiting time at the drive-through of a fast-food restaurant from the time an order is placed to the time the order is received is 84.7 seconds. A manager devises a new drive-through system that he believes will decrease wait time. As a test, he initiates the new system at his restaurant and measures the wait time for 10 randomly selected orders. The wait times are provided in the table to the right. Complete parts (a) and (b) below.

# Hypothesis Testing for Drive-Through Wait Times

The mean waiting time at a drive-through of a fast-food restaurant from the time an order is placed to the time the order is received is 84.7 seconds. A manager devises a new drive-through system that he believes will decrease wait time. As a test, he initiates the new system at his restaurant and measures the wait time for 10 randomly selected orders. The wait times are provided in the table to the right. Complete parts (a) and (b) below.

### Wait Times Data
| Time (sec) | Time (sec) |
|------------|------------|
| 105.3 | 79.0 |
| 66.4 | 95.7 |
| 56.7 | 84.7 |
| 75.0 | 71.1 |
| 66.5 | 79.6 |

## Part (a)
Because the sample size is small, the manager must verify that the wait time is normally distributed and the sample does not contain any outliers. The normal probability plot is shown below and the sample correlation coefficient is known to be $ r = 0.981 $.

**Question:** Are the conditions for testing the hypothesis satisfied?

**Answer:**
- **Yes**, the conditions *are* satisfied. The normal probability plot *is* linear enough, since the correlation coefficient is *greater* than the critical value. In addition, a boxplot does not show any outliers.

#### Normal Probability Plot
The normal probability plot depicted shows observed data versus expected data (z-scores). The data points primarily follow a straight line, indicating that the data is approximately normally distributed.

## Part (b)
**Question:** Is the new system effective? Conduct a hypothesis test using the P-value approach and a level of significance of $ \alpha = 0.01 $.

### Hypotheses
\[
H_0: \mu = 84.7
\]
\[
H_1: \mu < 84.7
\]

### Calculation
1. **Find the test statistic:**
\[
t_0 = 78 \quad \text{(Round to two decimal places as needed.)}
\]

2. **Find the P-value:**

The P-value is \( 13.73 \quad \text{(Round to three decimal places as needed

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

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