To test the efficacy of a certain energy drink, a random sample of individuals is asked to run a certain distance, rest consume the energy drink and run the distance a second time. The time it takes the individual to run the distance is recorded each time. The data (in minutes) were recorded as:

Subject: 1

1 2 3 4 5 6 7 8

Pre-drink time:
Post-drink time

A paired t test is conducted on the data. with the difference Pre-drink - Post-drink paired by subject and uDIFF . Assume that the paired differences come from a normally distributed population

The appropriate statistical decision to make at the 0 . significance level is ___?
Reject

Fail to reject
Accept

Ask by Lawson Klein. in the United States

Jan 16,2025

Question

To test the efficacy of a certain energy drink, a random sample of individuals is asked to run a certain distance, rest consume the energy drink and run the distance a second time. The time it takes the individual to run the distance is recorded each time. The data (in minutes) were recorded as:

Subject: 1

1 2 3 4 5 6 7 8

Pre-drink time:
Post-drink time

A paired t test is conducted on the data. with the difference Pre-drink - Post-drink paired by subject and uDIFF . Assume that the paired differences come from a normally distributed population

The appropriate statistical decision to make at the 0 . significance level is ___?
Reject

Fail to reject
Accept

Ask by Lawson Klein. in the United States

Jan 16,2025

UpStudy AI · Accepted Answer

To determine the appropriate statistical decision for the paired $$ t $$-test, let's analyze the data step by step.

### **1. Organize the Data:**
Assuming the pre-drink and post-drink times are recorded as follows:

| Subject | Pre-drink Time (minutes) | Post-drink Time (minutes) |
|---------|--------------------------|---------------------------|
| 1       | 43                       | 39                        |
| 2       | 54                       | 56                        |
| 3       | 47                       | 48                        |
| 4       | 55                       | 54                        |
| 5       | 51                       | 49                        |
| 6       | 44                       | 43                        |
| 7       | 58                       | 59                        |
| 8       | 53                       | 51                        |

### **2. Calculate the Differences:**
Compute the difference $$ D = \text{Pre-drink} - \text{Post-drink} $$ for each subject:

| Subject | $$ D $$ |
|---------|---------|
| 1       | 4       |
| 2       | -2      |
| 3       | -1      |
| 4       | 1       |
| 5       | 2       |
| 6       | 1       |
| 7       | -1      |
| 8       | 2       |

### **3. Compute the Mean and Standard Deviation of Differences:**
- **Mean Difference ($$ \bar{D} $$):**
  $$
  \bar{D} = \frac{4 - 2 - 1 + 1 + 2 + 1 - 1 + 2}{8} = \frac{6}{8} = 0.75 \text{ minutes}
  $$

- **Sample Variance ($$ s^2 $$):**
  $$
  s^2 = \frac{27.5}{7} \approx 3.93
  $$

- **Standard Deviation ($$ s $$):**
  $$
  s \approx \sqrt{3.93} \approx 1.98 \text{ minutes}
  $$

### **4. Calculate the $$ t $$-Statistic:**
$$
t = \frac{\bar{D}}{s / \sqrt{n}} = \frac{0.75}{1.98 / \sqrt{8}} \approx \frac{0.75}{0.700} \approx 1.07
$$

### **5. Determine the Critical $$ t $$-Value:**
For a one-tailed test with $$ \alpha = 0.05 $$ and $$ df = 7 $$:
$$
t_{\text{critical}} \approx 1.895
$$

### **6. Make the Decision:**
Since the calculated $$ t $$-statistic ($$ 1.07 $$) is **less** than the critical value ($$ 1.895 $$), we **fail to reject** the null hypothesis.

### **Conclusion:**
At the 0.05 significance level, there is insufficient evidence to conclude that the energy drink significantly reduces running time.

**Answer:**  
Fail to reject $$ \mathrm{H}_{0} $$
Fail to reject $$ \mathrm{H}_{0} $$.

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