The amount of oxygen consumption was measured in six individuals (n = 6) over two 10-minute periods while sitting with their eyes closed. During one period, they listened to an exciting adventure story; during the other, they heard restful music. The following are the results obtained:

Adventure Story: 6.1, 7.2, 5.7, 6.4, 5.8, 6.2

Music: 5.4, 6.7, 5.4, 6.2, 5.5, 6.0

Is there sufficient evidence to conclude that oxygen consumption is less when listening to the music versus the adventure story?

State the null and research (alternative) hypotheses in words and using symbols.  Conduct the appropriate hypothesis test with  = .01 and state your conclusion in terms of this problem.  Remember to use the 4 steps of hypothesis testing. Make sure to write APA style conclusions.

To determine if there is sufficient evidence to conclude that oxygen consumption is less when listening to music compared to an adventure story, we can conduct a paired sample t-test. Here’s how the hypothesis testing should be conducted:

### **Step 1: State the Hypotheses**

**Null Hypothesis (H₀):**
There is no difference in oxygen consumption between listening to restful music and an exciting adventure story.

In symbols:
\[
H_0: \mu_{\text{Music}} = \mu_{\text{Adventure Story}}
\]
or equivalently:
\[
H_0: \mu_D = 0
\]
where \(\mu_D\) represents the mean difference in oxygen consumption between the two conditions.

**Alternative Hypothesis (H₁):**
Oxygen consumption is less when listening to restful music compared to the exciting adventure story.

In symbols:
\[
H_1: \mu_{\text{Music}} < \mu_{\text{Adventure Story}}
\]
or equivalently:
\[
H_1: \mu_D < 0
\]

### **Step 2: Choose the Significance Level**
The significance level (\(\alpha\)) is given as 0.01.

### **Step 3: Conduct the Paired Sample t-Test**

First, calculate the differences in oxygen consumption between the two conditions for each individual.

\[
\text{Differences (D)} = \text{Adventure Story} - \text{Music}
\]
D: \(6.1 - 5.4\), \(7.2 - 6.7\), \(5.7 - 5.4\), \(6.4 - 6.2\), \(5.8 - 5.5\), \(6.2 - 6.0\)

\[
\text{Differences (D)}: 0.7, 0.5, 0.3, 0.2, 0.3, 0.2
\]

Next, calculate the mean difference (\(\bar{D}\)), standard deviation of the differences (SD), and then perform the t-test.

\[
\bar{D} = \frac{\sum D}{n} = \frac{0.7 + 0.5 + 0.3 + 0.2 + 0.3 + 0.2}{6}
\]
\[
\bar{D} = \frac{2.2}{6} = 0.367
\]

Now, calculate the standard deviation (SD) of the differences:
\[
\text{SD} = \sqrt{\frac{\sum (D_i - \bar{D})^2}{n - 1}}
\]

Finally, calculate the t-statistic:
\[
t = \frac{\bar{D}}{(\text{SD}/\sqrt{n})}
\]

### **Step 4: Make a Decision and State the Conclusion**

Compare the calculated t-value with the critical t-value from the t-distribution table with \(n-1 = 5\) degrees of freedom at \(\alpha = 0.01\) for a one-tailed test. If the calculated t-value is greater than the critical t-value, reject the null hypothesis.

Let me calculate these values for you to finalize the test.### **Step 3: Results of the Paired Sample t-Test**

- **Mean Difference (\(\bar{D}\))**: \(0.367\)
- **Standard Deviation of Differences (SD)**: \(0.197\)
- **Calculated t-statistic**: \(4.57\)
- **Critical t-value (one-tailed, \(\alpha = 0.01\))**: \(3.36\)
- **p-value**: \(0.997\)

### **Step 4: Decision and Conclusion**

Since the calculated t-statistic \(4.57\) is greater than the critical t-value \(3.36\), we reject the null hypothesis.

**APA Style Conclusion:**
There is sufficient evidence at the \(\alpha = 0.01\) level to conclude that oxygen consumption is significantly higher while listening to an exciting adventure story compared to restful music, \(t(5) = 4.57\), \(p < 0.01\).

---

Would you like more details or have any questions? Here are some related questions you might be interested in:

1. What would the results indicate if we used a different significance level, like 0.05?
2. How would the analysis differ if there were more than 6 participants?
3. Could the type of music or story affect the results, and how might that be tested?
4. What are the assumptions behind a paired sample t-test?
5. How would you conduct this analysis using a non-parametric test?

**Tip:** Always check the assumptions of normality and equal variances when performing t-tests to ensure the validity of your results.

Explore hypothesis testing to determine if oxygen consumption differs significantly between listening to music and an adventure story. Includes step-by-step t-test calculation, significance level, and conclusions at α = 0.01.

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