It Is Believed That Tai Chi Is An Ancient Chinese Practice

It Is Believed That Tai Chi An Ancient Chinese Practice Of Exercise A

It is believed that Tai Chi, an ancient Chinese practice of exercise and meditation, is more effective in relieving symptoms of chronic painful fibromyalgia than a widely-used wellness education method. To examine this belief, a study was conducted where 35 fibromyalgia patients were randomly assigned to participate in a 12-week Tai Chi class, and another 35 patients participated in a wellness education class. The outcomes showed that 27 patients in the Tai Chi group felt better at the end of the program, whereas 15 patients in the wellness education group reported feeling better. Let p1 represent the proportion of patients who feel better after Tai Chi, and p2 represent the proportion who feel better after wellness education. The significance level is 0.01.

Paper For Above instruction

The primary objective of this analysis is to determine whether Tai Chi is statistically more effective than wellness education in alleviating symptoms of fibromyalgia, based on the data collected from the two groups. This involves formulating hypotheses, conducting relevant significance tests using the p-value and critical value approaches, and interpreting the results. Additionally, a related question examines the implications of confidence intervals on estimation accuracy and reliability.

Formulation of Hypotheses

In hypothesis testing, the null hypothesis typically states that there is no difference between the two groups concerning the proportion who feel better. Therefore, the hypotheses can be formalized as follows:

  • Null hypothesis (H0): p1 = p2
  • Alternative hypothesis (Ha): p1 > p2

This setup indicates a one-tailed test, aiming to evaluate whether Tai Chi leads to a higher proportion of improved patients than wellness education.

Data Summary and Calculations

From the study, the data summarizes as:

  • Number of patients in Tai Chi group: n1 = 35
  • Number of patients feeling better in Tai Chi group: x1 = 27
  • Proportion feeling better in Tai Chi group: p1 = 27/35 ≈ 0.7714
  • Number of patients in Wellness Education group: n2 = 35
  • Number of patients feeling better in Wellness Education group: x2 = 15
  • Proportion feeling better in Wellness Education group: p2 = 15/35 ≈ 0.4286

The observed difference in sample proportions is approximately:

p1 - p2 ≈ 0.7714 - 0.4286 = 0.3428

P-value Approach

To perform the p-value approach, we compute the pooled sample proportion:

p̂pooled = (x1 + x2) / (n1 + n2) = (27 + 15) / (35 + 35) = 42 / 70 = 0.6

Calculate the standard error (SE) for the difference in proportions:

SE = √[p̂pooled(1 - p̂pooled)(1/n1 + 1/n2)]

SE = √[0.6 0.4 (1/35 + 1/35)] = √[0.24 (2/35)] = √[0.24 0.0571] ≈ √0.0137 ≈ 0.1172

Compute the z-statistic:

z = (p1 - p2) / SE = 0.3428 / 0.1172 ≈ 2.927

Using standard normal distribution tables or software, the p-value for z ≈ 2.927 (one-sided test) is approximately:

p-value ≈ 0.00175

Since p-value = 0.00175

Critical Value Approach

The critical value for a one-tailed test at α=0.01 is approximately 2.33 (from the standard normal distribution table). Our calculated z-value of approximately 2.927 exceeds this critical value (2.33). Therefore, we reject H0. This conclusion aligns with the p-value approach, indicating significant evidence favoring the effectiveness of Tai Chi over wellness education.

Comparison of the Conclusions

The consistency between the p-value approach and the critical value approach reinforces the robustness of the results. Both methods suggest that Tai Chi substantially improves fibromyalgia symptoms compared to wellness education at the 1% significance level.

Implications of Confidence Intervals on Estimating the Population Mean Age at Marriage

Moving on to the second part, the previous study provided a 95% confidence interval for the average age at marriage (μ) in the United States, ranging from 26.4 to 27.3 years based on a sample of 500 individuals. An independent study, using a separate sample, constructs its own 95% confidence interval for μ. The question pertains to how the width of this new interval compares to the previous one and what implications this has regarding the accuracy and likelihood of containing the true mean.

Analysis of Confidence Interval Widths

A confidence interval's width directly relates to the variability in the sample data and the sample size. Specifically, larger samples typically lead to narrower confidence intervals because of increased precision, assuming similar variability. Conversely, smaller samples often produce wider intervals due to greater variability and less precise estimates.

Given that the previous confidence interval for μ has a width of:

27.3 - 26.4 = 0.9 years

If the new confidence interval is narrower, it indicates higher precision, likely owed to a larger sample size or decreased variability. Conversely, a wider interval signals less precision or higher variability.

The Likely Scenario and Inference

Since the question states an independent sample is used, and assuming the new study is similar in methodology and variability, the second interval will typically be either narrower or wider depending on the sample size and data variability. Given the same confidence level (95%), the interval that is narrower would be more informative if it contains the same true mean, m, with same or higher likelihood. However, because the intervals are independent and constructed independently, the chance of each containing the true mean remains at 95%, regardless of their widths.

Thus, the best answer is:

  1. It may be wider or narrower, but it has the same chance to contain m.

Conclusion

In conclusion, statistical hypothesis testing plays a vital role in evaluating the effectiveness of various health interventions, such as Tai Chi, by leveraging sample data to make inferences about the population. The combined use of p-value and critical value approaches provides a comprehensive understanding of the results, with the consistency between these methods reinforcing the reliability of the findings. Furthermore, understanding confidence intervals and their relationship with sample size and variability enhances our ability to interpret and compare estimates across different studies, emphasizing the importance of methodological rigor and appropriate data collection in statistical analysis.

References

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  • Li, J., et al. (2012). Tai Chi and depression: A meta-analysis of randomized controlled trials. Journal of Psychiatric Research, 46(4), 577-582.
  • Smith, M. (2020). Understanding confidence intervals: Basics and applications. Statistica Sinica, 30(2), 567-586.
  • U.S. Census Bureau. (2022). Marital status and age at first marriage: 2020. Retrieved from https://www.census.gov
  • Cummings, S. R., et al. (2020). Sample size determination and confidence interval estimation in clinical research. Statistics in Medicine, 39(15), 2320-2335.
  • Casella, G., & Berger, R. L. (2002). Statistical Inference (2nd ed.). Duxbury Press.
  • Higgins, J., & Green, S. (2011). Cochrane Handbook for Systematic Reviews of Interventions. The Cochrane Collaboration.
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  • Newcombe, R. G. (1998). Two-sided confidence intervals for the single proportion: Comparison of seven methods. Statistics in Medicine, 17(8), 857–872.
  • Zhang, W., et al. (2010). Effectiveness of Tai Chi for treating osteoarthritis of the knee: A systematic review. Rheumatology, 49(2), 330–434.

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