Chapter 9, Problem 3CS

To determine

Find the 95% confidence interval for the group means or group proportions for the health characteristics.

Answer to Problem 3CS

The 95% confidence intervals for the each of the characteristics are as 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	$\underset{\_}{\left(-0.04895,0.03585\right)}$

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.

95% confidence interval for the variable “Age”:

Software Procedure:

Step by step procedure to find 95% confidence interval forthe difference between the two population means using the MINITAB software:

• Choose Stat > Basic Statistics > 2-Sample t.
• Choose Summarized data.
• In first sample, enter the sample size as 731, sample mean as 64 and sample standard deviation as 11.
• In second sample, enter the sample size as 1,089, sample mean as 65 and sample standard deviation as 11.
• Choose Options.
• In Confidence level, enter 95.
• Enter 0 in Hypothesized value.
• In Alternative, select not equal to
• Click OK in all the dialogue boxes.

Output using the MINITAB software is given below:

From, the MINITAB output the 95% confidence interval for the difference between the mean age of new treatment and standard treatment is $\underset{\_}{\left(-2.032,0.032\right)}$.

95% confidence interval for the variable “Systolic blood pressure”:

Software Procedure:

Step by step procedure to find 95% confidence interval forthe difference between the two population means using the MINITAB software:

• Choose Stat > Basic Statistics > 2-Sample t.
• Choose Summarized data.
• In first sample, enter the sample size as 731, sample mean as 121 and sample standard deviation as 18.
• In second sample, enter the sample size as 1,089, sample mean as 124 and sample standard deviation as 17.
• Choose Options.
• In Confidence level, enter 95.
• Enter 0 in Hypothesized value.
• In Alternative, select not equal to
• Click OK in all the dialogue boxes.

Output using the MINITAB software is given below:

From, the MINITAB output the 95% confidence interval for the difference between the mean systolic blood pressure of new treatment and standard treatmentis $\underset{\_}{\left(-4.651,-1.349\right)}$.

95% confidence interval for the variable “Diastolic blood pressure”:

Software Procedure:

Step by step procedure to find 95% confidence interval for the difference between the two population means using the MINITAB software:

• Choose Stat > Basic Statistics > 2-Sample t.
• Choose Summarized data.
• In first sample, enter the sample size as 731, sample mean as 71 and sample standard deviation as 10.
• In second sample, enter the sample size as 1,089, sample mean as 72 and sample standard deviation as 10.
• Choose Options.
• In Confidence level, enter 95.
• Enter 0 in Hypothesized value.
• In Alternative, select not equal to
• Click OK in all the dialogue boxes.

Output using the MINITAB software is given below:

From, the MINITAB output the 95% confidence interval for the difference between the mean diastolic blood pressure of new treatment and standard treatmentis $\underset{\_}{\left(-1.938,-0.062\right)}$.

95% confidence interval for the variable “Treatment for hypertension”:

The sample size of standard treatment is $n_1=731$. The percentage with the characteristics “Treatment for hypertension under standard treatment” is $p_1=0.632$.

Hence, the specified characteristics of treatment for hypertension under new treatment is $x_1=n_1\times p_1=461.992$.

The sample size of new treatment is $n_2=1,089$. The percentage with the characteristics “Treatment for hypertension under new treatment” is $p_2=0.637$.

Hence, the specified characteristics of treatment for hypertension under new treatment is $x_2=n_2\times p_2=693.693$.

Software Procedure:

Step by step procedure to find 95% confidence interval for the difference between the two population proportions using the MINITAB software:

• Choose Stat > Basic Statistics > 2-Proportions.
• Choose Summarized data.
• In first sample, enter the number of trials as 731 and number of events as 462.
• In second sample, enter the number of trials as 1,089 and number of events as 694.
• Choose Options.
• In Confidence level, enter 95.
• Enter 0 in Hypothesized value.
• In Alternative, select not equal to
• Click OK in all the dialogue boxes.

Output using the MINITAB software is given below:

From, the MINITAB output the 95% confidence interval for the difference between the proportion of treatment for hypertension under new treatment and standard treatmentis $\underset{\_}{\left(-0.0504,0.0399\right)}$.

95% confidence interval for the variable “Atrial fibrillation”:

The sample size of standard treatment is $n_1=731$. The percentage with the characteristics “Atrial fibrillation under standard treatment” is $p_1=0.126$.

Hence, the specified characteristics of treatment for Atrial fibrillation under standard treatment is $x_1=n_1\times p_1=92.106$.

The sample size of new treatment is $n_2=1,089$. The percentage with the characteristics “Atrial fibrillation under new treatment” is $p_2=0.111$.

Hence, the specified characteristics of Atrial fibrillation under new treatment is $x_2=n_2\times p_2=120.879$.

Software Procedure:

Step by step procedure to find 95% confidence interval for the difference between the two populationproportions using the MINITAB software:

• Choose Stat > Basic Statistics > 2-Proportions.
• Choose Summarized data.
• In first sample, enter the number of trials as 731 and number of events as 92.
• In second sample, enter the number of trials as 1,089 and number of events as 121.
• Choose Options.
• In Confidence level, enter 95.
• Enter 0 in Hypothesized value.
• In Alternative, select not equal to
• Click OK in all the dialogue boxes.

Output using the MINITAB software is given below:

From, the MINITAB output the 95% confidence interval for the difference between the proportion of Atrial fibrillation under new treatment and standard treatment is $\underset{\_}{\left(-0.0157,0.0452\right)}$.

95% confidence interval for the variable “Diabetes”:

The sample size of standard treatment is $n_1=731$. The percentage with the characteristics “Diabetes under standard treatment” is $p_1=0.302$.

Hence, the specified characteristics of diabetes under standard treatment is $x_1=n_1\times p_1=220.762$.

The sample size of new treatment is $n_2=1,089$. The percentage with the characteristics “Diabetes under new treatment” is $p_2=0.306$.

Hence, the specified characteristics of diabetes under new treatment is $x_2=n_2\times p_2=333.234$.

Software Procedure:

Step by step procedure to find 95% confidence interval for the difference between the two population proportions using the MINITAB software:

• Choose Stat > Basic Statistics > 2-Proportions.
• Choose Summarized data.
• In first sample, enter the number of trials as 731 and number of events as 221.
• In second sample, enter the number of trials as 1,089 and number of events as 333.
• Choose Options.
• In Confidence level, enter 95.
• Enter 0 in Hypothesized value.
• In Alternative, select not equal to
• Click OK in all the dialogue boxes.

Output using the MINITAB software is given below:

From, the MINITAB output the95% confidence interval for the difference between the proportion of Diabetes under new treatment and standard treatment is $\underset{\_}{\left(-0.0467,0.0396\right)}$.

95% confidence interval for the variable “Cigarette smoking”:

The sample size of standard treatment is $n_1=731$. The percentage with the characteristics “Cigarette smoking under standard treatment” is $p_1=0.128$.

Hence, the specified characteristics of cigarette smoking under standard treatment is $x_1=n_1\times p_1=93.568$.

The sample size of new treatment is $n_2=1,089$. The percentage with the characteristics “Cigarette smoking under new treatment” is $p_2=0.114$.

Hence, the specified characteristics of cigarette smoking under new treatment is $x_2=n_2\times p_2=124.146$.

Software Procedure:

Step by step procedure to find 95% confidence interval for the difference between the two population proportions using the MINITAB software:

• Choose Stat > Basic Statistics > 2-Proportions.
• Choose Summarized data.
• In first sample, enter the number of trials as 731 and number of events as 94.
• In second sample, enter the number of trials as 1,089 and number of events as 124.
• Choose Options.
• In Confidence level, enter 95.
• Enter 0 in Hypothesized value.
• In Alternative, select not equal to
• Click OK in all the dialogue boxes.

Output using the MINITAB software is given below:

From, the MINITAB output the 95% confidence interval for the difference between the proportion of Cigarette smoking under new treatment and standard treatment is $\underset{\_}{\left(-0.016,0.0455\right)}$.

95% confidence interval for the variable “Coronary bypass surgery”:

The sample size of standard treatment is $n_1=731$. The percentage with the characteristics “Coronary bypass surgery under standard treatment” is $p_1=0.285$.

Hence, the specified characteristics of Coronary bypass surgery under standard treatment is $x_1=n_1\times p_1=208.335$.

The sample size of new treatment is $n_2=1,089$. The percentage with the characteristics “Coronary bypass surgery under new treatment” is $p_2=0.291$.

Hence, the specified characteristics of coronary bypass surgery under new treatment is $x_2=n_2\times p_2=316.899$.

Software Procedure:

Step by step procedure to find 95% confidence interval for the difference between the two population proportions using the MINITAB software:

• Choose Stat > Basic Statistics > 2-Proportions.
• Choose Summarized data.
• In first sample, enter the number of trials as 1,089 and number of events as 317.
• In second sample, enter the number of trials as 731 and number of events as 208.
• Choose Options.
• In Confidence level, enter 95.
• Enter 0 in Hypothesized value.
• In Alternative, select not equal to
• Click OK in all the dialogue boxes.

Output using the MINITAB software is given below:

From, the MINITAB output the 95% confidence interval for the difference between the proportion of Coronary bypass surgery under new treatment and standard treatment is $\underset{\_}{\left(-0.04895,0.03585\right)}$.