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Effect of Aspirin on Blood Clotting Blood clotting is due to a sequence of chemical reactions. The protein thrombin initiates blood clotting by working with another protein, prothrombin. It is common to measure an individual's blood clotting time as prothrombin time, the time between the start of the thrombinprothrombin reaction and the formation of the clot. Researchers wanted to study the effect of aspirin on prothrombin time. They randomly selected 12 subjects and measured the prothrombin time (in seconds) without taking aspirin and 3 hours after taking two aspirin tablets. They obtained the following data. Does the evidence suggest that aspirin affects the median time it takes for a clot to form? Use the \(\alpha=0.05\) level of significance. $$ \begin{array}{ccc|ccc} & \text { Before } & \text { After } & & \text { Before } & \text { After } \\\ \text { Subject } & \text { Aspirin } & \text { Aspirin } & \text { Subject } & \text { Aspirin } & \text { Aspirin } \\ \hline 1 & 12.3 & 12.0 & 7 & 11.3 & 10.3 \\ \hline 2 & 12.0 & 12.3 & 8 & 11.8 & 11.3 \\ \hline 3 & 12.0 & 12.5 & 9 & 11.5 & 11.5 \\ \hline 4 & 13.0 & 12.0 & 10 & 11.0 & 11.5 \\ \hline 5 & 13.0 & 13.0 & 11 & 11.0 & 11.0 \\ \hline 6 & 12.5 & 12.5 & 12 & 11.3 & 11.5 \end{array} $$

Step by step solution

01

Formulate the Hypotheses

State the null hypothesis (H_0) and the alternative hypothesis (H_1). H_0: The median prothrombin time is the same before and after taking aspirin. H_1: The median prothrombin time is different before and after taking aspirin.

02

Choose the Appropriate Test

Since the sample size is small (n = 12) and we are comparing two related samples, use the Wilcoxon signed-rank test to evaluate the hypotheses.

03

Calculate the Differences

Calculate the differences between the prothrombin times before and after taking aspirin for each subject. 1: 12.3 - 12.0 = 0.32: 12.0 - 12.3 = -0.33: 12.0 - 12.5 = -0.54: 13.0 - 12.0 = 1.05: 13.0 - 13.0 = 0.06: 12.5 - 12.5 = 0.07: 11.3 - 10.3 = 1.08: 11.8 - 11.3 = 0.59: 11.5 - 11.5 = 0.010: 11.0 - 11.5 = -0.511: 11.0 - 11.0 = 0.012: 11.3 - 11.5 = -0.2

04

Rank the Absolute Differences

Rank the absolute values of the non-zero differences from smallest to largest, ignoring the signs. 0.2 (rank 1), 0.3 (rank 2.5), 0.3 (rank 2.5), 0.5 (rank 5.5), 0.5 (rank 5.5), 1.0 (rank 8.5), 1.0 (rank 8.5)

05

Assign Signs to Ranks

Assign the appropriate sign to each rank based on the original difference. 0.3 (rank 2.5), -0.3 (rank -2.5), -0.5 (rank -5.5), 1.0 (rank 8.5), 0.5 (rank 5.5), -0.5 (rank -5.5), -0.2 (rank -1), 1.0 (rank 8.5)

06

Calculate the Test Statistic

Sum the ranks of the positive differences (T_+) and the ranks of the negative differences (T_-). T_+ = 2.5 + 8.5 + 5.5 + 8.5 = 25T_- = -2.5 - 5.5 - 5.5 - 1 = -14.5

07

Determine the Critical Value

For the given significance level a = 0.05 and n = 12, find the critical value from the Wilcoxon signed-rank test table. For this example, the critical value at a = 0.05 is 10.

08

Make the Decision

Compare the test statistic to the critical value. If T_+ <= 10 or T_- <= 10, reject the null hypothesis. In this case, T_+ = 25, which is greater than 10, and T_- = -14.5, which is not less than -10. Thus, we do not reject the null hypothesis.

09

State the Conclusion

Based on the Wilcoxon signed-rank test at a = 0.05, there is no significant evidence to conclude that aspirin affects the median prothrombin time.

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Key Concepts

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

Hypothesis Testing

In the context of statistical analysis, hypothesis testing is used to determine whether there is enough evidence to reject a null hypothesis. We start with a null hypothesis, denoted as H_0, which represents a default position that there is no effect or no difference. The alternative hypothesis, denoted as H_1, represents what we want to prove is true.
In this exercise, the null hypothesis (H_0) is that the median prothrombin time is the same before and after taking aspirin. The alternative hypothesis (H_1) is that the median prothrombin time is different before and after taking aspirin.
Through hypothesis testing, we statistically evaluate these hypotheses using collected data. The significance level, \( \alpha = 0.05 \), is a threshold for determining whether to reject H_0. If the test statistic falls within the critical value range, we can confidently reject H_0 and accept H_1.

Prothrombin Time

Prothrombin time measures how long it takes for blood to clot after the activation of prothrombin. This time is crucial for assessing blood clotting function.
In this scenario, researchers measured the prothrombin time of subjects before and after taking aspirin. Aspirin is known for its blood-thinning properties, which could potentially affect prothrombin time.
This exercise focuses on comparing these times to see if aspirin significantly changes the clotting time. Accurate measurement and analysis are essential to understand aspirin's impact and make informed medical decisions.

Paired Sample

A paired sample involves measurements from the same subjects under two different conditions. This method increases the reliability of findings because it controls for individual differences.
In our exercise, each subject's prothrombin time was measured twice: before taking aspirin and after taking aspirin. This provides paired data for each subject, allowing for a direct comparison of conditions.
Using such a paired sample design helps isolate the effect of aspirin from other variables, enhancing the validity and accuracy of the results.