Chapter 5.3, Problem 3E

Find the level of a two-sided confidence interval that is based on the given value of $t_{n-1,\alpha /2}$ and the given sample size.

1. a. t = 2.776, sample size 5.
2. b. t = 2.718, sample size 12.
3. c. t = 5.841, sample size 4.
4. d. t = 1.325, sample size 21.
5. e. t = 1.746, sample size 17.

To determine

Find the level of a two-sided confidence interval that is based on the given value of $t_{n-1,\frac{\alpha }{2}}$ for $t=2.776$, sample size 5.

Answer to Problem 3E

The level of a two-sided confidence interval that is based on the given value of $t_{n-1,\frac{\alpha }{2}}$ for $t=2.776$, sample size 5 is 95%.

Explanation of Solution

Calculation:

Degrees of freedom:

$\begin{array}{c}n-1=5-1\\ =4\end{array}$

Software Procedure:

Step-by-step procedure to obtain the level of significance using the MINITAB software:

• Choose Graph > Probability Distribution Plot choose View Probability> OK.
• From Distribution, choose ‘t distribution.
• In Degrees of freedom, enter 4.
• Click the Shaded Area tab.
• Choose X Value and Two Tail for the region of the curve to shade.
• Enter the X value as 2.776.
• Click OK.

Output using the MINITAB software is given below:

From the output,

$\begin{array}{c}\text{Level}=100\left(1-\alpha \right)\%\\ =100\left(1-0.05\right)\%\\ =95\%\end{array}$

Thus, the level of a two-sided confidence interval that is based on the given value of $t_{n-1,\frac{\alpha }{2}}$ for $t=2.776$, sample size 5 is 95%.

To determine

Find the level of a two-sided confidence interval that is based on the given value of $t_{n-1,\frac{\alpha }{2}}$ for $t=2.718$, sample size 12.

Answer to Problem 3E

The level of a two-sided confidence interval that is based on the given value of $t_{n-1,\frac{\alpha }{2}}$ for $t=2.718$, sample size 12 is 98%.

Explanation of Solution

Calculation:

Degrees of freedom:

$\begin{array}{c}n-1=12-1\\ =11\end{array}$

Software Procedure:

Step-by-step procedure to obtain the level of significance using the MINITAB software:

• Choose Graph > Probability Distribution Plot choose View Probability> OK.
• From Distribution, choose ‘t distribution.
• In Degrees of freedom, enter 11.
• Click the Shaded Area tab.
• Choose X Value and Two Tail for the region of the curve to shade.
• Enter the X value as 2.718.
• Click OK.

Output using the MINITAB software is given below:

From the output,

$\begin{array}{c}\text{Level}=100\left(1-\alpha \right)\%\\ =100\left(1-0.02\right)\%\\ =98\%\end{array}$

Thus, the level of a two-sided confidence interval that is based on the given value of $t_{n-1,\frac{\alpha }{2}}$ for $t=2.776$, sample size 5 is 98%.

To determine

Find the level of a two-sided confidence interval that is based on the given value of $t_{n-1,\frac{\alpha }{2}}$ for $t=5.841$, sample size 4.

Answer to Problem 3E

The level of a two-sided confidence interval that is based on the given value of $t_{n-1,\frac{\alpha }{2}}$ for $t=5.841$, sample size 4 is 99%.

Explanation of Solution

Calculation:

Degrees of freedom:

$\begin{array}{c}n-1=4-1\\ =3\end{array}$

Software Procedure:

Step-by-step procedure to obtain the level of significance using the MINITAB software:

• Choose Graph > Probability Distribution Plot choose View Probability> OK.
• From Distribution, choose ‘t distribution.
• In Degrees of freedom, enter 3.
• Click the Shaded Area tab.
• Choose X Value and Two Tail for the region of the curve to shade.
• Enter the X value as 5.841.
• Click OK.

Output using the MINITAB software is given below:

From the output,

$\begin{array}{c}\text{Level}=100\left(1-\alpha \right)\%\\ =100\left(1-0.01\right)\%\\ =99\%\end{array}$

From the output, the level of a two-sided confidence interval that is based on the given value of $t_{n-1,\frac{\alpha }{2}}$ for $t=5.841$, sample size 4 is 99%.

To determine

Find the level of a two-sided confidence interval that is based on the given value of $t_{n-1,\frac{\alpha }{2}}$ for $t=1.325$, sample size 21.

Answer to Problem 3E

The level of a two-sided confidence interval that is based on the given value of $t_{n-1,\frac{\alpha }{2}}$ for $t=1.325$, sample size 21 is 80%.

Explanation of Solution

Calculation:

Degrees of freedom:

$\begin{array}{c}n-1=21-1\\ =20\end{array}$

Software Procedure:

Step-by-step procedure to obtain the level of significance using the MINITAB software:

• Choose Graph > Probability Distribution Plot choose View Probability> OK.
• From Distribution, choose ‘t distribution.
• In Degrees of freedom, enter 20.
• Click the Shaded Area tab.
• Choose X Value and Two Tail for the region of the curve to shade.
• Enter the X value as 1.325.
• Click OK.

Output using the MINITAB software is given below:

From the output,

$\begin{array}{c}\text{Level}=100\left(1-\alpha \right)\%\\ =100\left(1-0.2\right)\%\\ =80\%\end{array}$

From the output, the level of a two-sided confidence interval that is based on the given value of $t_{n-1,\frac{\alpha }{2}}$ for $t=1.325$, sample size 21 is 80%.

To determine

Find the level of a two-sided confidence interval that is based on the given value of $t_{n-1,\frac{\alpha }{2}}$ for $t=1.746$, sample size 17.

Answer to Problem 3E

The level of a two-sided confidence interval that is based on the given value of $t_{n-1,\frac{\alpha }{2}}$ for $t=1.746$, sample size 17 is 90%.

Explanation of Solution

Calculation:

Degrees of freedom:

$\begin{array}{c}n-1=17-1\\ =16\end{array}$

Software Procedure:

Step-by-step procedure to obtain the level of significance using the MINITAB software:

• Choose Graph > Probability Distribution Plot choose View Probability> OK.
• From Distribution, choose ‘t distribution.
• In Degrees of freedom, enter 16.
• Click the Shaded Area tab.
• Choose X Value and Two Tail for the region of the curve to shade.
• Enter the X value as 1.746.
• Click OK.

Output using the MINITAB software is given below:

From the output,

$\begin{array}{c}\text{Level}=100\left(1-\alpha \right)\%\\ =100\left(1-0.10\right)\%\\ =90\%\end{array}$

Thus, the level of a two-sided confidence interval that is based on the given value of $t_{n-1,\frac{\alpha }{2}}$ for $t=1.746$, sample size 17 is 90%.