This Problem Set Will Practice Solving Problems ✓ Solved

This problem set will give you practice in solving problems relating to statistical inference and hypothesis testing learned in this module

This problem set will give you practice in solving problems relating to statistical inference and hypothesis testing learned in this module. Problems will be similar to those you will face on the quiz in Module Six and will include one or two real-world applications to prepare you to think like a biostatistician. To complete this assignment, review the Module Four Problem Set document. Please complete all problems and show any work necessary. Document with problems.

Sample Paper For Above instruction

Introduction

Statistical inference and hypothesis testing are fundamental components of biostatistics that facilitate decision-making based on data. This problem set aims to provide practical exercises that reinforce these concepts through real-world applications, similar to what students will encounter in Module Six assessments. By engaging with these problems, students will develop the skills necessary to analyze data critically and draw valid conclusions in biological research contexts.

Problem 1: Understanding Hypotheses and Test Statistics

A researcher hypothesizes that a new drug reduces blood pressure more effectively than the standard medication. The average reduction in blood pressure with the standard drug is known to be 8 mm Hg, with a standard deviation of 2 mm Hg. A sample of 30 patients using the new drug shows an average reduction of 9.2 mm Hg. Conduct a hypothesis test at the 0.05 significance level to determine if the new drug is more effective.

Solution:

Null hypothesis (H0): μ = 8 mm Hg

Alternative hypothesis (H1): μ > 8 mm Hg

Given data: sample mean (x̄) = 9.2 mm Hg, population standard deviation (σ) = 2 mm Hg, sample size (n) = 30

Calculate the z-test statistic:

z = (x̄ - μ₀) / (σ / √n) = (9.2 - 8) / (2 / √30) ≈ 1.2 / (2 / 5.477) ≈ 1.2 / 0.365 ≈ 3.29

Critical value at α = 0.05 for a one-tailed test: z ≈ 1.645

The calculated z-value (3.29) exceeds 1.645, so we reject H0. There is sufficient evidence to conclude that the new drug reduces blood pressure more effectively than the standard medication.

Problem 2: Real-World Application – Comparing Two Treatments

A biostatistician wants to compare the effectiveness of two different diets on weight loss. Group A (n=50) follows Diet A and loses an average of 5 kg with a standard deviation of 1.5 kg. Group B (n=50) follows Diet B and loses an average of 4.2 kg with a standard deviation of 1.8 kg. Conduct a two-sample t-test at the 0.05 level to determine if there is a significant difference in mean weight loss between the two diets.

Solution:

Formulate hypotheses:

H0: μA - μB = 0

H1: μA - μB ≠ 0

Calculate the pooled standard error and t-statistic:

t = (x̄₁ - x̄₂) / √(s₁²/n₁ + s₂²/n₂) = (5 - 4.2) / √( (1.5)²/50 + (1.8)²/50 ) ≈ 0.8 / √(0.045 + 0.065) ≈ 0.8 / √0.11 ≈ 0.8 / 0.332 ≈ 2.41

Degrees of freedom approximate to 98. The critical t-value at α=0.05 for a two-tailed test is approximately 1.984.

Since 2.41 > 1.984, we reject H0, indicating a significant difference in weight loss between the two diets.

Conclusion

These exercises demonstrate essential applications of hypothesis testing in biostatistics, including z-tests and t-tests, relevant to real-world health research. Accurately formulating hypotheses, calculating the relevant test statistics, and comparing against critical values are crucial steps in making evidence-based decisions. Developing proficiency in these methods enables researchers to draw valid conclusions from their data, ultimately contributing to advancements in healthcare and disease management.

References

  • Gelman, A., & Hill, J. (2006). Data Analysis Using Regression and Multilevel/Hierarchical Models. Cambridge University Press.
  • Field, A. (2013). Discovering Statistics Using R. Sage Publications.
  • Moore, D. S., McCabe, G. P., & Craig, B. A. (2017). Introduction to the Practice of Statistics. W. H. Freeman.
  • Ott, R. L., & Longnecker, M. (2015). An Introduction to Statistical Methods and Data Analysis. Cengage Learning.
  • Daniel, W. W. (2010). Biostatistics: A Foundation for Analysis in the Health Sciences. John Wiley & Sons.
  • Rothman, K. J., Greenland, S., & Lash, T. L. (2008). Modern Epidemiology. Lippincott Williams & Wilkins.
  • Weiss, N. A. (2012). Introductory Statistics. Pearson.
  • McDonald, J. H. (2014). Handbook of Biological Statistics. Sparky House Publishing.
  • Lauritzen, S. L. (2017). Statistical Modelling. Oxford University Press.
  • Hastie, T., Tibshirani, R., & Friedman, J. (2009). The Elements of Statistical Learning. Springer.

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