Chapter 8.2, Problem 11E

For a t distribution with 16 degrees of freedom, find the area, or probability, in each region.

1. a. To the right of 2.120
2. b. To the left of 1.337
3. c. To the left of −1.746
4. d. To the right of 2.583
5. e. Between −2.120 and 2.120
6. f. Between −1.746 and 1.746

To determine

Find the probability to the right of 2.120 with 16 degrees of freedom.

Answer to Problem 11E

The probability to the right of 2.120 with 16 degrees of freedom is 0.025.

Explanation of Solution

Calculation:

The probability to right of 2.120 is $P\left(x>2.120\right)$.

Software procedure:

Step-by-step procedure to obtain the probability value using Excel:

• Open an EXCEL sheet and select the cell A1.
• Enter the formula =T.DIST.RT(2.12,16) in the cell A1.
• Press Enter.

Output obtained using EXCEL software is given below:

From the output, the value of $P\left(x>2.120\right)$ is 0.025.

To determine

Find the probability to the left of 1.337 with 16 degrees of freedom.

Answer to Problem 11E

The probability to the left of 1.337 with 16 degrees of freedom is 0.90.

Explanation of Solution

Calculation:

The probability to the left of 1.337 is $P\left(x\le 1.337\right)=1-P\left(x>1.337\right)$.

Software Procedure:

Step-by-step procedure to obtain the probability value using Excel:

• Open an EXCEL sheet and select the cell A1.
• Enter the formula =T.DIST.RT(1.337,16) in the cell A1.
• Press Enter.

Output obtained using EXCEL software is given below:

From the output:

$\begin{array}{c}P\left(x>1.337\right)=0.10\\ P\left(x\le 1.337\right)=1-P\left(x>1.337\right)\\ =1-0.10\\ =0.90\end{array}$.

Thus, the value of $P\left(x\le 1.337\right)$ is 0.90.

To determine

Find the probability to the left of –1.746 with 16 degrees of freedom.

Answer to Problem 11E

The probability to the left of –1.746 with 16 degrees of freedom is 0.05.

Explanation of Solution

Calculation:

The probability to the left of –1.746 is $P\left(x\le -1.746\right)=1-P\left(x>-1.746\right)$.

Software Procedure:

Step-by-step procedure to obtain the probability value using Excel:

• Open an EXCEL sheet and select the cell A1.
• Enter the formula =T.DIST.RT(–1.746,16) in the cell A1.
• Press Enter.

Output obtained using EXCEL software is given below:

From the output:

$\begin{array}{c}P\left(x>-1.746\right)=0.95\\ P\left(x\le -1.746\right)=1-P\left(x>-1.746\right)\\ =1-0.95\\ =0.05\end{array}$.

Thus, the value of $P\left(x\le -1.746\right)$ is 0.05.

To determine

Find the probability to the right of 2.583 with 16 degrees of freedom.

Answer to Problem 11E

The probability to the right of 2.583 with 16 degrees of freedom is 0.01.

Explanation of Solution

Calculation:

The probability of right of 2.583 is, $P\left(x>2.583\right)$.

Software Procedure:

Step-by-step procedure to obtain the probability value using Excel:

• Open an EXCEL sheet and select the cell A1.
• Enter the formula =T.DIST.RT(2.583,16) in the cell A1.
• Press Enter.

Output obtained using EXCEL software is given below:

From the output, the value of $P\left(x>2.583\right)$ is 0.01.

To determine

Find the probability between –2.120 and 2.120 with 16 degrees of freedom.

Answer to Problem 11E

The probability between –2.120 and 2.120 with 16 degrees of freedom is 0.95.

Explanation of Solution

Calculation:

The probability between –2.120 and 2.120 is $P\left(-2.120\le x\le 2.120\right)$.

Software Procedure:

Step-by-step procedure to obtain the probability value using Excel:

• Open an EXCEL sheet and select the cell A1.
• Enter the formula =T.DIST.RT(2.120,16) in the cell A1.
• Press Enter.

Output obtained using EXCEL software is given below:

From the output, the value of $P\left(x>2.120\right)$ is 0.025.

The probability between –2.120 and 2.120 is as follows:

$\begin{array}{c}P\left(-2.120\le x\le 2.120\right)=1-2P\left(x>2.120\right)\\ =1-2\left(0.025\right)\\ =0.95\end{array}$.

Thus, the value of $P\left(-2.120\le x\le 2.120\right)$ is 0.95.

To determine

Find the probability between –1.746 and 1.746 with 16 degrees of freedom.

Answer to Problem 11E

The probability between –1.746 and 1.746 with 16 degrees of freedom is 0.90.

Explanation of Solution

Calculation:

The probability between –1.746 and 1.746 is $P\left(-1.746\le x\le 1.746\right)$.

Software Procedure:

Step-by-step procedure to obtain the probability value using Excel:

• Open an EXCEL sheet and select the cell A1.
• Enter the formula =T.DIST.RT(1.746,16) in the cell A1.
• Press Enter.

Output obtained using EXCEL software is given below:

From the output, the value of $P\left(x>1.746\right)$ is 0.05.

The probability between –1.746 and 1.746 is as follows:

$\begin{array}{c}P\left(-1.746\le x\le 1.746\right)=1-2P\left(x>1.746\right)\\ =1-2\left(0.05\right)\\ =0.90\end{array}$.

Thus, the value of $P\left(-1.746\le x\le 1.746\right)$ is 0.90.