Question

Eight people were asked to perform a simple puzzle-assembly task under normal and stressful conditions. The stressful time consisted of a stimulus delivered to subjects every 30 seconds until the task was completed. Blood pressure readings taken under both conditions are given in the table. Do the data present sufficient evidence to indicate higher blood pressure readings under stressful conditions? Analyze the data using the Wilcoxon signed-rank test for a paired experiment. $$ \begin{array}{ccc} \hline \text { Subject } & \text { Normal } & \text { Stressful } \\ \hline 1 & 126 & 130 \\ 2 & 117 & 118 \\ 3 & 115 & 125 \\ 4 & 118 & 120 \\ 5 & 118 & 121 \\ 6 & 128 & 125 \\ 7 & 125 & 130 \\ 8 & 120 & 120 \\ \hline \end{array} $$

Short answer

Answer: No, there is not enough evidence to conclude that blood pressure readings are significantly higher under stressful conditions compared to normal conditions.

Step-by-step solution

Calculate the Differences Between Paired Observations

First, we need to calculate the differences between the paired observations for each subject. We will take the blood pressure reading under stressful conditions and subtract the reading under normal conditions. $$ \begin{array}{cccc} \hline \text{Subject} & \text{Normal} & \text{Stressful} & \text{Difference (Stressful - Normal)} \\ \hline 1 & 126 & 130 & 4 \\ 2 & 117 & 118 & 1 \\ 3 & 115 & 125 & 10 \\ 4 & 118 & 120 & 2 \\ 5 & 118 & 121 & 3 \\ 6 & 128 & 125 & -3 \\ 7 & 125 & 130 & 5 \\ 8 & 120 & 120 & 0 \\ \hline \end{array} $$

Rank the Absolute Differences and Assign Signs

Next, we will rank the absolute differences between the paired readings, ignoring the 0 difference. Then, we will assign the corresponding signs from the original differences. $$ \begin{array}{cccccc} \hline \text{Subject} & \text{Difference} & | \text{Difference} | & \text{Rank} & \text{Sign} & \text{Signed Rank} \\ \hline 1 & 4 & 4 & 3 & + & +3 \\ 2 & 1 & 1 & 1 & + & +1 \\ 3 & 10 & 10 & 7 & + & +7 \\ 4 & 2 & 2 & 2 & + & +2 \\ 5 & 3 & 3 & 2.5 & + & +2.5 \\ 6 & -3 & 3 & 2.5 & - & -2.5 \\ 7 & 5 & 5 & 4 & + & +4 \\ 8 & 0 & 0 & - & - & - \\ \hline \end{array} $$

Calculate T-Statistics

Now we will calculate the T-statistics. T-positive (T+) is the sum of the positive signed ranks, and T-negative (T-) is the sum of the negative signed ranks. $$ T+ = +1 + 2 + 2.5 + 3 + 4 + 7 = 19.5 \\ T- = |-2.5| = 2.5 $$

Determine the Test Statistic and Critical Value

The test statistic (T) is the smaller of the two T-values, so T = 2.5. We will now determine the critical value using a Wilcoxon signed-rank test table. Since we have 7 non-zero differences (n=7), and assuming a significance level of 0.05 (which is commonly used), we find that the critical value (T_0) is 2.

Compare Test Statistic to Critical Value and Draw Conclusion

Finally, we will compare our test statistic (T = 2.5) to the critical value (T_0 = 2). Since T > T_0, we fail to reject the null hypothesis. This means that we do not have enough evidence to conclude that the blood pressure readings under stressful conditions are significantly higher than those under normal conditions, as per the Wilcoxon signed-rank test.

Key concepts

Paired Experiment Analysis

When researchers are interested in examining the impact of a specific treatment or condition on the same individuals, a paired experiment plays a crucial role in the statistical analysis. This design involves taking multiple measurements on the same subject, such as before and after applying a treatment or under different conditions. For instance, analyzing blood pressure readings under normal and stressful conditions for the same individuals, offers insights into how stress influences cardiovascular health.

In paired experiments, the primary focus is on the difference between pairs of observations. These differences are pivotal as they aim to isolate and evaluate the effect of the treatment or condition. Statistically, this analysis minimizes the variance by accounting for innate subject variability, offering a more precise estimate of the treatment effect. It's vital to perform such experiments under controlled conditions to ensure that the observed differences are attributable solely to the treatment effect and not to external variability.

Non-Parametric Statistics

In the statistical world, data doesn't always conform to the assumptions of traditional (parametric) statistical tests. That's where non-parametric statistics come into play. These tests, such as the Wilcoxon signed-rank test, do not assume a specific distribution for the data, making them highly versatile and robust for analyses.

Non-parametric tests are especially useful when data are ordinal, when their level of measurement is ranked or not normally distributed. In the exercise, the Wilcoxon signed-rank test is employed, which is perfect for analyzing the differences in paired observations without assuming the data come from a normal distribution. This test calculates ranks of the differences and uses these ranks to assess whether the two conditions (normal and stressful in our case) demonstrate significantly different effects.

Blood Pressure Readings Comparison

Comparing blood pressure readings under different conditions can provide valuable health-related insights. Blood pressure is a critical health indicator and monitoring its response to stress can inform about potential heart-related conditions or the general wellbeing of individuals. In our exercise, the comparison of readings under normal versus stressful conditions aims to identify whether stress has a measurable impact on blood pressure.

The comparison involves a set of methodological steps: calculating the paired differences, ranking these differences in terms of their absolute values while retaining their signs, and then using a statistical test like the Wilcoxon signed-rank test to determine if the observed differences are significant. Although the test found no significant difference in our case, it's a robust and insightful way to analyze related measurements on the same subjects. When applied correctly, it can reveal both subtle and pronounced effects of conditions like stress on physiological measures.