91Ó°ÊÓ

Does exercise help blood pressure? Exercise 10.50 in Chapter 10 discussed a pilot study of people who suffer from abnormally high blood pressure. 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 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. Show how to analyze the data with the sign test. State the hypotheses, find the P-value, and interpret. $$ \begin{aligned} &\begin{array}{ccc} \hline \text { Subject } & \text { Before } & \text { After } \\ \hline 1 & 150 & 130 \\ 2 & 165 & 140 \\ 3 & 135 & 120 \\ \hline \end{array}\\\ &\text { 4. More on blood pressure } \quad \text { Refer to the previous } \end{aligned} $$

Step by step solution

01

State the Hypotheses

We need to determine if the exercise program has an effect on blood pressure. Thus, we set our hypotheses as follows: \(H_0: \) There is no effect of exercise on blood pressure (i.e., there is no difference in before and after blood pressure readings). \(H_a: \) Exercise leads to a decrease in blood pressure (i.e., after measurements are lower than before). This is a one-tailed test since we are only concerned with a decrease.

02

Calculate Differences and Determine Signs

Calculate the difference for each subject by subtracting the after measurement from the before measurement: Subject 1: 150 - 130 = 20 (positive difference), Subject 2: 165 - 140 = 25 (positive difference), Subject 3: 135 - 120 = 15 (positive difference). All differences are positive, suggesting a decrease in blood pressure for each subject.

03

Conduct the Sign Test

In our sign test, we observe the number of positive and negative differences. Here, all differences are positive (3 positives, 0 negatives). We use this count to conduct the test. With our alternative hypothesis being a decrease, we focus on the positives.

04

Calculate the P-value

For small samples like this, the sign test can reference the binomial distribution. With 3 positives out of 3 trials and under the null hypothesis (chance of decrease is 0.5), the P-value is calculated as: \[ P(X \geq 3) = P(X = 3) = \left( \frac{1}{2} \right)^3 = 0.125. \]

05

Interpret the Results

The P-value is 0.125, which is above the common significance level of 0.05. Thus, we fail to reject the null hypothesis. There is not enough statistical evidence to conclusively say that exercise lowers blood pressure in this sample.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Hypothesis Testing

Hypothesis testing is a fundamental method in statistics used to make decisions or inferences about a population based on a sample. It involves setting up two competing statements or hypotheses. The null hypothesis ( H_0 ) typically states that there is no effect or difference, while the alternative hypothesis ( H_a ) suggests the opposite. In the context of the blood pressure exercise, the null hypothesis states that walking briskly does not affect systolic blood pressure (no difference between before and after measurements). On the other hand, the alternative hypothesis suggests that walking leads to a decrease in blood pressure (after readings are lower than before).

Hypothesis testing involves:

• Setting Hypotheses: Defining H_0 and H_a clearly.
• Choosing the Test: Selecting an appropriate test based on sample size and data type. In this case, a sign test is chosen.
• Calculating the Test Statistic: Including the difference or signs observed in the sample.
• P-value Calculation: Determining the probability under the null hypothesis.
• Decision Making: Based on the P-value, deciding whether to reject the null hypothesis.

Understanding these steps helps in critically assessing the effectiveness of interventions like exercise on health parameters.

Systolic Blood Pressure

Systolic blood pressure is the higher of the two numbers in a blood pressure reading and signifies the pressure in your arteries when your heart beats. It is a critical measure because high systolic pressure can increase the risk of cardiovascular diseases.

When conducting experiments to see how factors like exercise influence blood pressure, focusing on systolic pressure allows us to assess potential improvements in heart health. The data collected in any studies involving systolic blood pressure, such as in the exercise with the three patients, provides insights into the heart’s condition before and after interventions.

Key points include:

• Measurement: It's essential to ensure accurate readings both before and after interventions like exercise.
• Significance in Studies: Systolic pressure changes are often observed in research because even slight improvements can reduce the risk of significant health issues.

Different factors such as stress, diet, and physical activity can influence these readings, which is why controlled experiments help isolate the effect of one factor, like exercise.

P-value Interpretation

The P-value is a crucial result in hypothesis testing that indicates the strength of evidence against the null hypothesis. It represents the probability of observing the test results, or something more extreme, assuming the null hypothesis is true.

In simpler terms:

• A small P-value (typically ≤ 0.05) indicates strong evidence against the null and suggests rejecting H_0 , supporting the presence of an effect.
• A large P-value (greater than 0.05) indicates weak evidence against H_0 . In this exercise on systolic blood pressure, the P-value was calculated at 0.125.
• Since 0.125 > 0.05, it means we do not have sufficient evidence to reject H_0 .

P-value interpretation helps in understanding the implications of statistical findings. However, it’s also important to consider practical significance, not just statistical significance. In the context of real-life applications like health interventions, even if statistical evidence is weak, noticeable health improvements can be of clinical importance.