Question

At Rachel’s \(11th\) birthday party, eight girls were timed to see how long (in seconds) they could hold their breath in a relaxed position. After a two-minute rest, they timed themselves while jumping. The girls thought that the mean difference between their jumping and relaxed times would be zero. Test their hypothesis.

Short answer

Considering the table 10.24 the data consists of matched samples,

1. The null hypothesis is given by, \(H_{0}:\mu_{d}=0\)

2. Alternate hypothesis is given by, \(H_{a}:\mu_{d}\neq 0\)

3. The random variable \(X_{d}\) is the average difference between jumping and relaxed times.

4. Student’s t distribution with \(7\) degree of freedom

5. The paired test static for the mean difference is,

After ENTER all values the OUTPUT shown below,

6. The \(p-value\) from the output is \(0.175\)

7. Using the information from previous exercise to sketch a picture of this situation.

Clearly label and scale the horizontal axis, and shade the graph the region(s) corresponding to the \(p-\)value.

8. \(\alpha=0.05\)

Decision: Null hypothesis is not rejected.

\(p-value>alpha\)

Conclusion: At the \(5%\) level of significance, there is insufficient evidence to conclude that the average difference is not zero.

Step-by-step solution

Step 1. Given information

Given table,

Step 2. Calculation

Considering the table 10.24 the data consists of matched samples,

1. The null hypothesis is given by, \(H_{0}:\mu_{d}=0\)

2. Alternate hypothesis is given by, \(H_{a}:\mu_{d}\neq 0\)

3. The random variable \(X_{d}\) is the average difference between jumping and relaxed times.

4. Student’s t distribution with \(7\) degree of freedom

5. The paired test static for the mean difference is,

After ENTER all values the OUTPUT shown below,

6. The \(p-value\) from the output is \(0.175\)

7. Using the information from previous exercise to sketch a picture of this situation.

Clearly label and scale the horizontal axis, and shade the graph the region(s) corresponding to the \(p-\)value.

8. \(\alpha=0.05\)

Decision: Null hypothesis is not rejected.

\(p-value>alpha\)

Conclusion: At the \(5%\) level of significance, there is insufficient evidence to conclude that the average difference is not zero.