Chapter 9, Problem 4CS

To determine

Find the health characteristics for which the assignment to groups is not balanced using the confidence interval obtained in Exercise-3.

Answer to Problem 4CS

The assignments to the groups are not balanced for the health characteristics “Systolic blood pressure” and “Diastolic blood pressure”.

Explanation of Solution

Calculation:

The data represents the means and standard deviations corresponding to three characteristics of standard treatment and new treatment.

Furthermore, the data corresponding to the percentage with the characteristics for the 5 characteristics for standard treatment and new treatment is given.

Here, it is given that the sample size of standard treatment is $n_1=731$ and the sample size of new treatment is $n_2=1,089$.

${\overline{x}}_i\text{'}s$ are the sample means of each characteristics and $s_i\text{'}s$ are the sample standard deviations of each characteristics. ${\mu }_i\text{'}s$ be the population means of each characteristics.

Hypothesis test for the variable “Age”:

The hypotheses are given below:

Null hypothesis:

$H_0:{\mu }_1={\mu }_2$

That is, there is no significant difference between the mean age of new treatment and standard treatment.

Alternate hypothesis:

$H_1:{\mu }_1\ne {\mu }_2$

That is, there is a significant difference between the mean age of new treatment and standard treatment.

Hypothesis test for the variable “Systolic blood pressure”:

The hypotheses are given below:

Null hypothesis:

$H_0:{\mu }_1={\mu }_2$

That is, there is no significant difference between the mean systolic blood pressure of new treatment and standard treatment.

Alternate hypothesis:

$H_1:{\mu }_1\ne {\mu }_2$

That is, there is a significant difference between the mean systolic blood pressure of new treatment and standard treatment.

Hypothesis test for the variable “Diastolic blood pressure”:

The hypotheses are given below:

Null hypothesis:

$H_0:{\mu }_1={\mu }_2$

That is, there is no significant difference between the meandiastolic blood pressure of new treatment and standard treatment.

Alternate hypothesis:

$H_1:{\mu }_1\ne {\mu }_2$

That is, there is a significant difference between the meandiastolic blood pressure of new treatment and standard treatment.

Hypothesis test for the variable “Treatment for hypertension”:

The hypotheses are given below:

Null hypothesis:

$H_0:p_1=p_2$

That is, there is no significant difference between the proportion of treatment for hypertension under new treatment and standard treatment.

Alternate hypothesis:

$H_1:p_1\ne p_2$

That is, there is a significant difference between the proportion of treatment for hypertension under new treatment and standard treatment.

Hypothesis test for the variable “Atrial fibrillation”:

The hypotheses are given below:

Null hypothesis:

$H_0:p_1=p_2$

That is, there is no significant difference between the proportion of Atrial fibrillation under new treatment and standard treatment.

Alternate hypothesis:

$H_1:p_1\ne p_2$

That is, there is a significant difference between the proportion of Atrial fibrillation under new treatment and standard treatment.

Hypothesis test for the variable “Diabetes”:

The hypotheses are given below:

Null hypothesis:

$H_0:p_1=p_2$

That is, there is no significant difference between the proportion of Diabetes under new treatment and standard treatment.

Alternate hypothesis:

$H_1:p_1\ne p_2$

That is, there is a significant difference between the proportion of Diabetes under new treatment and standard treatment.

Hypothesis test for the variable “Cigarette smoking”:

The hypotheses are given below:

Null hypothesis:

$H_0:p_1=p_2$

That is, there is no significant difference between the proportion of Cigarette smoking under new treatment and standard treatment.

Alternate hypothesis:

$H_1:p_1\ne p_2$

That is, there is a significant difference between the proportion of Cigarette smoking under new treatment and standard treatment.

Hypothesis test for the variable “Coronary bypass surgery”:

The hypotheses are given below:

Null hypothesis:

$H_0:p_1=p_2$

That is, there is no significant difference between the proportion of Coronary bypass surgery under new treatment and standard treatment.

Alternate hypothesis:

$H_1:p_1\ne p_2$

That is, there is a significant difference between the proportion of Coronary bypass surgery under new treatment and standard treatment.

From Exercise-3, the confidence interval for the each test areas follows:

Characteristic	95% confidence interval
Age	$\underset{\_}{\left(-2.032,0.032\right)}$
Systolic blood pressure	$\underset{\_}{\left(-4.651,-1.349\right)}$
Diastolic blood pressure	$\underset{\_}{\left(-1.938,-0.062\right)}$
Treatment for hypertension	$\underset{\_}{\left(-0.0504,0.0399\right)}$
Atrial fibrillation	$\underset{\_}{\left(-0.0157,0.0452\right)}$
Diabetes	$\underset{\_}{\left(-0.0467,0.0396\right)}$
Cigarette smoking	$\underset{\_}{\left(-0.016,0.0455\right)}$
Coronary bypass surgery	$\underset{\_}{\left(-0.04895,0.03585\right)}$

Decision in the given scenario:

If “0” lies between the confidence interval, then fail to reject the null hypothesis $H_0$.

If “0” does not lie between the confidence interval, then reject the null hypothesis $H_0$.

Conclusion based on confidence intervals:

Characteristic	P–value	Whether or not “0” lies in the confidence interval	Acceptance of rejection of null hypothesis
Age	$\underset{\_}{\left(-2.032,0.032\right)}$	“0” lies in the confidence interval	Fail to reject the null hypothesis
Systolic blood pressure	$\underset{\_}{\left(-4.651,-1.349\right)}$	“0” does not lie in the confidence interval	Reject the null hypothesis
Diastolic blood pressure	$\underset{\_}{\left(-1.938,-0.062\right)}$	“0” does not lie in the confidence interval	Reject the null hypothesis
Treatment for hypertension	$\underset{\_}{\left(-0.0504,0.0399\right)}$	“0” lies in the confidence interval	Fail to reject the null hypothesis
Atrial fibrillation	$\underset{\_}{\left(-0.0157,0.0452\right)}$	“0” lies in the confidence interval	Fail to reject the null hypothesis
Diabetes	$\underset{\_}{\left(-0.0467,0.0396\right)}$	“0” lies in the confidence interval	Fail to reject the null hypothesis
Cigarette smoking	$\underset{\_}{\left(-0.016,0.0455\right)}$	“0” lies in the confidence interval	Fail to reject the null hypothesis
Coronary bypass surgery	$\underset{\_}{\left(-0.04895,0.03585\right)}$	“0” lies in the confidence interval	Fail to reject the null hypothesis

Conclusions:

There is no significant difference between the mean age of new treatment and standard treatment.

There is a significant difference between the mean systolic blood pressure of new treatment and standard treatment.

There is a significant difference between the mean diastolic blood pressure of new treatment and standard treatment.

There is no significant difference between the proportion of treatment for hypertension under new treatment and standard treatment.

There is no significant difference between the proportion of Atrial fibrillation under new treatment and standard treatment.

There is no significant difference between the proportion of cigarette smoking under new treatment and standard treatment.

There is no significant difference between the proportion of coronary bypass surgery under new treatment and standard treatment.

From this it is known that, the assignments to the groups are not balanced for the health characteristics “Systolic blood pressure” and “Diastolic blood pressure”.

Thus, the assignments to the groups are not balanced for the health characteristics “Systolic blood pressure” and “Diastolic blood pressure”.