91Ó°ÊÓ

Several recent studies have suggested that people who suffer from abnormally high blood pressure can benefit from regular exercise. A medical researcher decides to test her belief that walking briskly for at least half an hour a day has the effect of lowering blood pressure. She conducts a small pilot study. If results from it are supportive, she will apply for funding for a larger study. She randomly samples three of her patients who have high blood pressure. She measures their systolic blood pressure initially and then again a month later after they participate in her exercise program. The table shows the results. $$ \begin{array}{ccc} \hline \text { Subject } & \text { Before } & \text { After } \\ \hline 1 & 150 & 130 \\ 2 & 165 & 140 \\ 3 & 135 & 120 \\ \hline \end{array} $$ a. Explain why the three before observations and the three after observations are dependent samples. b. Find the sample mean of the before scores, the sample mean of the after scores, and the sample mean of \(d=\) before \(-\) after. How are they related? c. Find a \(95 \%\) confidence interval for the difference between the population means of subjects before and after going through such a study. Interpret.

Step by step solution

01

Understanding Dependent Samples

The before and after observations are dependent samples because they involve the same subjects. For each participant, their blood pressure was measured twice—once before and once after the exercise program. This creates a paired dataset where each "before" observation is directly related to its corresponding "after" observation for each subject.

02

Calculating Sample Means

First, calculate the sample mean of the before scores: \(\text{Mean}_{\text{before}} = \frac{150 + 165 + 135}{3} = 150\). Next, calculate the sample mean of the after scores: \(\text{Mean}_{\text{after}} = \frac{130 + 140 + 120}{3} = 130\). To find the sample mean of the differences, calculate each difference for the subjects: \(d_1 = 150 - 130 = 20\), \(d_2 = 165 - 140 = 25\), \(d_3 = 135 - 120 = 15\). The mean of these differences is \(\text{Mean}_d = \frac{20 + 25 + 15}{3} = 20\). This mean difference relates the other means as \(\text{Mean}_{\text{before}} - \text{Mean}_{\text{after}} = \text{Mean}_d\).

03

Calculating the Confidence Interval

To find a 95% confidence interval for the mean difference, calculate the standard deviation of the differences: \(s_d = \sqrt{\frac{(20-20)^2 + (25-20)^2 + (15-20)^2}{2}} = \sqrt{\frac{0 + 25 + 25}{2}} = 5\). The standard error is \(SE_d = \frac{5}{\sqrt{3}} \approx 2.89\). The critical value for a 95% confidence interval with 2 degrees of freedom is approximately 4.303. Compute the margin of error: \(ME = 4.303 \times 2.89 \approx 12.44\). The confidence interval is \(20 \pm 12.44\), which is \(7.56, 32.44\). This interval suggests that the exercise program likely reduces systolic blood pressure by 7.56 to 32.44 points, on average.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Dependent Samples

In medical studies, especially those involving treatment outcomes, dependent samples are quite common. In this case, dependent samples arise when observations are made on the same subjects under different conditions. Here, the exercise involved measuring blood pressure for the same individuals, both before and after participating in a physical activity regimen. This means each subject's measurements are linked because they provide two observations from the same individual.

• Why they are dependent: Since each subject provides both a 'before' and an 'after' measurement, any change detected is directly attributable to the individual’s reaction to the treatment.
• Advantages: Removes variability that occurs between different participants, given that each person serves as their own control. This often leads to more powerful statistical tests.

Understanding that samples are dependent helps in choosing appropriate statistical methods, ensuring accurate interpretation of the results.

Confidence Interval

A confidence interval is an essential statistical tool used to estimate the range within which we believe a true population parameter will fall. Here, we aim to estimate the true mean difference in blood pressure before and after exercise.

• What it signifies: The 95% confidence interval gives us two numbers between which we expect the true mean difference to lie, covering 95% of such intervals constructed in this way from repeated samples.
• Importance in medicine: Confidence intervals provide a range of plausible values for an effect size (like the impact of exercise on blood pressure), giving researchers valuable insights that go beyond single-point estimates.

The confidence interval calculated in the exercise suggests that, with 95% certainty, the true mean reduction in systolic blood pressure due to exercise is between 7.56 and 32.44 points.

Paired Data

Paired data arise when each observational unit, such as a patient, provides two data points. This is evident in the exercise study, where each subject has a pair of blood pressure readings (before and after exercise).

• Characteristics: Unlike independent groups, paired datasets involve related measurements. Here, each subject's 'before' datum aligns directly with their 'after' datum, creating pairs of observations.
• Statistical handling: Analysis methods for paired data, like the paired t-test, directly compare these pairs to assess the effect of treatments, controlling for individual baselines.

By focusing on the differences between paired observations, researchers can increase the sensitivity of the test, identifying changes or effects that might remain hidden in unpaired data analysis.

Sample Mean

The sample mean gives a central measure of tendency for the dataset, offering an average value around which data points are distributed. For the blood pressure study, we calculated multiple sample means: before and after exercise, and for the differences between these points.

• How it's computed: Add all the data points together and divide by the total number of observations. It's a simple yet powerful summary of the dataset.
• Role in paired comparisons: The mean of differences (before minus after) provides an overall measure of the treatment’s impact. It aids in understanding whether the intervention had a general effect across all subjects.

The sample mean of the differences (\[\text{Mean}_d = 20\]), illustrates how much, on average, the exercise lowered blood pressure, reinforcing its utility in measuring treatment effects in paired studies.