Question

A study was conducted to determine whether relaxation training, aided by biofeedback and meditation, could help in reducing high blood pressure. Subjects were randomly allocated to a biofeedback group or a control group. The biofeedback group received training for 8 weeks. The table reports the reduction in systolic blood pressure $(\mathrm{mm}\mathrm{Hg})$ after eight weeks. ${}^{42}[$ Note: Formula $(6.7.1)$ yields 190 degrees of freedom for these data. (a) Construct a $95\mathrm{\%}$ confidence interval for the difference in mean response. (b) Interpret the confidence interval from part (a) in the context of this setting. $$\begin{array}{|lcc|}\hline & \text{ Biofeedback }& \text{ Control }\\ n& 99& 93\\ \overline{y}& 13.8& 4.0\\ \text{ SE }& 1.34& 1.30\\ \hline\end{array}$$

Short answer

The 95% confidence interval for the difference in mean reduction of systolic blood pressure is approximately $(9.42,10.18)$ mm Hg, meaning the true mean reduction for the biofeedback group is likely between $9.42$ mm Hg and $10.18$ mm Hg greater than the control group.

Step-by-step solution

Define the Variables

Let's define the sample sizes, means, and standard errors for the two groups: for the biofeedback group, we have a sample size of $n_1=99$, the mean reduction in blood pressure is ${\overline{y}}_1=13.8$ mm Hg, and the standard error is $S{E}_1=1.34$ mm Hg. For the control group, the sample size is $n_2=93$, the mean reduction is ${\overline{y}}_2=4.0$ mm Hg, and the standard error is $S{E}_2=1.30$ mm Hg.

Calculate the Standard Error for the Difference in Means

The standard error for the difference in means between two independent samples is given by the formula $S{E}_{difference}=\sqrt{S{E}_1^2/n_1+S{E}_2^2/n_2}$. Plug in the values to get $S{E}_{difference}=\sqrt{{1.34}^2/99+{1.30}^2/93}$.

Calculate the Standard Error for the Difference

Performing the calculations for the previous step gives $S{E}_{difference}=\sqrt{{1.34}^2/99+{1.30}^2/93}=\sqrt{0.018149+0.018182}=\sqrt{0.036331}=0.190605$ mm Hg.

Determine the Confidence Interval

The 95% confidence interval for the difference in mean response is calculated as ${\overline{y}}_1-{\overline{y}}_2\pm t_{\alpha /2}\cdot S{E}_{difference}$. We use the provided degrees of freedom and a t-distribution table or calculator to find the appropriate $t_{\alpha /2}$ value for 190 degrees of freedom, which is approximately 1.97 for a two-tailed test.

Calculate the Confidence Interval

Calculate the 95% confidence interval using the mean differences and the calculated standard error: $CI=(13.8-4.0)\pm 1.97\cdot 0.190605$. This results in $CI=9.8\pm 0.3753915$, which gives the intervals $9.8-0.3753915$ and $9.8+0.3753915$.

Compute the Interval Bounds

The lower bound of the confidence interval is $9.8-0.3753915=9.4246085$ mm Hg and the upper bound is $9.8+0.3753915=10.1753915$ mm Hg. This results in a 95% confidence interval for the difference in mean response of approximately $9.42$ mm Hg to $10.18$ mm Hg.

Interpret the Confidence Interval

The 95% confidence interval $(9.42,10.18)$ mm Hg suggests that we can be 95% confident that the true mean reduction in systolic blood pressure for the biofeedback group exceeds that of the control group by between $9.42$ mm Hg and $10.18$ mm Hg after eight weeks.

Key concepts

Biofeedback and Relaxation Training

Biofeedback and relaxation techniques are therapeutic interventions designed to help individuals improve their health by using signals from their own bodies. In the context of hypertension, biofeedback and relaxation training aim to reduce high systolic blood pressure, which is the pressure in the arteries when the heart beats. This approach may involve methods such as guided imagery, deep breathing exercises, or mindfulness meditation, often complemented by technology that gives real-time feedback on physiological functions like heart rate and muscle tension.

Studies have shown that these techniques can help in lowering blood pressure, likely due to their effects on reducing stress and promoting relaxation of the vascular system. The use of biofeedback can empower individuals to gain control over certain bodily processes that are typically autonomic, which may contribute to managing conditions like hypertension.

Systolic Blood Pressure Reduction

Systolic blood pressure, the upper number in a blood pressure reading, indicates the force that blood exerts on the artery walls during heartbeats. High systolic blood pressure, or hypertension, can lead to severe health issues like heart disease, stroke, and kidney problems. Reducing systolic blood pressure is thus a critical objective in managing cardiovascular health.

Effective reduction can result from a variety of interventions, including medication, lifestyle changes, and as highlighted in the exercise, relaxation techniques. In the given study, the participants in the biofeedback group exhibited a mean reduction in systolic blood pressure, indicating the potential effectiveness of non-pharmaceutical methods in managing blood pressure levels.

Standard Error for Difference in Means

The standard error for the difference in means is a statistical measure used to express the variability of the difference between two sample means. In practical terms, this measure helps us determine how much the difference in sample means (such as the reduction in systolic blood pressure between two groups) deviates from the actual difference in population means.

The formula for calculating the standard error for the difference in means is $S{E}_{difference}=\sqrt{\frac{S{E}_1^2}{n_1}+\frac{S{E}_2^2}{n_2}}$. It accounts for the standard errors of each group and their respective sample sizes. The smaller the standard error, the more precisely we can estimate the true difference between the two populations. Correct calculation of this error is vital for constructing accurate confidence intervals.

Interpretation of Confidence Intervals

A confidence interval is a range of values within which we can be a certain percentage confident that the true value lies. For instance, a 95% confidence interval suggests that if we were to take 100 different samples and compute a confidence interval for each, we expect about 95 of them to contain the true mean difference between populations.

Interpreting our confidence interval in the context of this study, we are 95% confident that the mean reduction in systolic blood pressure due to biofeedback and relaxation training lies between 9.42 mm Hg and 10.18 mm Hg greater than the control group. This interval does not mean that the actual mean difference will be within this range with absolute certainty. Instead, it means there's a high level of confidence that the range includes the true mean difference. This interpretation aids clinicians and researchers in making informed decisions about the effectiveness of interventions like biofeedback in reducing systolic blood pressure.