Chapter 9, Problem 107CE

a.

To determine

Find the value of $\beta$ risk if the true means are 495, 490, 485 and 480.

Answer to Problem 107CE

The $\beta$ risk values are tabulated below:

True mean	$\beta$ risk
495	0.7915
490	0.4913
485	0.1962
480	0.0457

Explanation of Solution

Calculation:

The given information is that the sample of size 4 are considered. The know standard deviation is 12. Here, the researcher interested to test the hypothesis that $\mu =500$.

The test hypotheses are given below:

Null hypothesis:

$H_0:\mu \ge 500$.

Alternative hypothesis:

$H_1:\mu <500$.

If the true mean is 495, the value of $\beta$ risk is calculated as follows:

Software procedure:

Step-by-step software procedure to obtain power using MINITAB is as follows:

• Select Stat > Power and sample size > 1 sample z.
• Enter Sample sizes as 4, Differences as –5, and Standard deviation as 12.
• Click OK.
• Output using MINITAB software is given below:

From the output, the power is 0.2085.

The value of $\beta$ risk is,

$\begin{array}{c}\beta =1-\text{Power}\\ =1-0.2085\\ =0.7915\end{array}$

Thus, the value of $\beta$ risk is 0.7915.

If the true mean is 490, the value of $\beta$ risk is calculated as follows:

Software procedure:

Step-by-step software procedure to obtain power using MINITAB is as follows:

• Select Stat > Power and sample size > 1 sample z.
• Enter Sample sizes as 4, Differences as –10, and Standard deviation as 12.
• Click OK.
• Output using MINITAB software is given below:

From the output, the power is 0.5087.

The value of $\beta$ risk is,

$\begin{array}{c}\beta =1-\text{Power}\\ =1-0.5087\\ =0.4913\end{array}$

Thus, the value of $\beta$ risk is 0.4913.

If the true mean is 485, the value of $\beta$ risk is calculated as follows:

Software procedure:

Step-by-step software procedure to obtain power using MINITAB is as follows:

• Select Stat > Power and sample size > 1 sample z.
• Enter Sample sizes as 4, Differences as –15, and Standard deviation as 12.
• Click OK.
• Output using MINITAB software is given below:

From the output, the power is 0.8038.

The value of $\beta$ risk is,

$\begin{array}{c}\beta =1-\text{Power}\\ =1-0.8038\\ =0.1962\end{array}$

Thus, the value of $\beta$ risk is 0.1962.

If the true mean is 480, the value of $\beta$ risk is calculated as follows:

Software procedure:

Step-by-step software procedure to obtain power using MINITAB is as follows:

• Select Stat > Power and sample size > 1 sample z.
• Enter Sample sizes as 4, Differences as –20, and Standard deviation as 12.
• Click OK.
• Output using MINITAB software is given below:

From the output, the power is 0.9543.

The value of $\beta$ risk is,

$\begin{array}{c}\beta =1-\text{Power}\\ =1-0.9543\\ =0.0457\end{array}$

Thus, the value of $\beta$ risk is 0.0457.

b.

To determine

Find the power if the true means are 495, 490, 485 and 480 and sketch the power curve.

Answer to Problem 107CE

The power values are tabulated below:

True mean	Power
495	0.2085
490	0.5087
485	0.8038
480	0.9543

• Output using MINITAB software is given below:

Explanation of Solution

Calculation:

If the true mean is 495, the power is calculated as follows:

Software procedure:

Step-by-step software procedure to obtain power using MINITAB is as follows:

• Select Stat > Power and sample size > 1 sample z.
• Enter Sample sizes as 4, Differences as –5, and Standard deviation as 12.
• Click OK.
• Output using MINITAB software is given below:

From the output, the power is 0.2085.

If the true mean is 490, the power is calculated as follows:

Software procedure:

Step-by-step software procedure to obtain power using MINITAB is as follows:

• Select Stat > Power and sample size > 1 sample z.
• Enter Sample sizes as 4, Differences as –10, and Standard deviation as 12.
• Click OK.
• Output using MINITAB software is given below:

From the output, the power is 0.5087.

If the true mean is 485, the power is calculated as follows:

Software procedure:

Step-by-step software procedure to obtain power using MINITAB is as follows:

• Select Stat > Power and sample size > 1 sample z.
• Enter Sample sizes as 4, Differences as –15, and Standard deviation as 12.
• Click OK.
• Output using MINITAB software is given below:

From the output, the power is 0.8038.

If the true mean is 480, the power is calculated as follows:

Software procedure:

Step-by-step software procedure to obtain power using MINITAB is as follows:

• Select Stat > Power and sample size > 1 sample z.
• Enter Sample sizes as 4, Differences as –20, and Standard deviation as 12.
• Click OK.
• Output using MINITAB software is given below:

From the output, the power is 0.9543.

Sketch the power curve.

Software procedure:

Step-by-step software procedure to construct the power curve using MINITAB is as follows:

• Choose Graph > Scatterplot.
• Choose With Connect Line, and then click OK.
• Under Y variables, enter Power.
• Under X variables, enter True mean.
• Click OK.

c.

To determine

Find the $\beta$ risk and power using $n=16$ and also sketch the power curve.

Answer to Problem 107CE

The $\beta$ risk and power values are tabulated below:

True mean	Power	$\beta$ risk
495	0.5087	0.4913
490	0.9543	0.0457
485	0.9996	0.0004
480	1	0

• Output using MINITAB software is given below:

Explanation of Solution

Calculation:

The given information is that the sample of size 16 are considered. The know standard deviation is 12. Here, the researcher interested to test the hypothesis that $\mu =500$.

The true means are 495, 490, 485 and 480.

If the true mean is 495, the $\beta$ risk and power is calculated as follows:

Software procedure:

Step-by-step software procedure to obtain power using MINITAB is as follows:

• Select Stat > Power and sample size > 1 sample z.
• Enter Sample sizes as 16, Differences as –5, and Standard deviation as 12.
• Click OK.
• Output using MINITAB software is given below:

From the output, the power is 0.5087.

The value of $\beta$ risk is,

$\begin{array}{c}\beta =1-\text{Power}\\ =1-0.5087\\ =0.4913\end{array}$

Thus, the value of $\beta$ risk is 0.4913.

If the true mean is 490, the $\beta$ risk and power is calculated as follows:

Software procedure:

Step-by-step software procedure to obtain power using MINITAB is as follows:

• Select Stat > Power and sample size > 1 sample z.
• Enter Sample sizes as 16, Differences as –10, and Standard deviation as 12.
• Click OK.
• Output using MINITAB software is given below:

From the output, the power is 0.9543.

The value of $\beta$ risk is,

$\begin{array}{c}\beta =1-\text{Power}\\ =1-0.9543\\ =0.0457\end{array}$

Thus, the value of $\beta$ risk is 0.0457.

If the true mean is 485, the $\beta$ risk and power is calculated as follows:

Software procedure:

Step-by-step software procedure to obtain power using MINITAB is as follows:

• Select Stat > Power and sample size > 1 sample z.
• Enter Sample sizes as 16, Differences as –15, and Standard deviation as 12.
• Click OK.
• Output using MINITAB software is given below:

From the output, the power is 0.9996.

The value of $\beta$ risk is,

$\begin{array}{c}\beta =1-\text{Power}\\ =1-0.9996\\ =0.0004\end{array}$

Thus, the value of $\beta$ risk is 0.0004.

If the true mean is 480, the $\beta$ risk and power is calculated as follows:

Software procedure:

Step-by-step software procedure to obtain power using MINITAB is as follows:

• Select Stat > Power and sample size > 1 sample z.
• Enter Sample sizes as 16, Differences as –20, and Standard deviation as 12.
• Click OK.
• Output using MINITAB software is given below:

From the output, the power is 1.

The value of $\beta$ risk is,

$\begin{array}{c}\beta =1-\text{Power}\\ =1-1\\ =0\end{array}$

Thus, the value of $\beta$ risk is 0.

Sketch the power curve.

Software procedure:

Step-by-step software procedure to construct the power curve using MINITAB is as follows:

• Choose Graph > Scatterplot.
• Choose With Connect Line, and then click OK.
• Under Y variables, enter Power.
• Under X variables, enter True mean.
• Click OK.