Solution Of STAT 3704 HW 5 Question 1: 4.12 A And B

Solution of STAT 3704 HW 5 Question 1: 4.12 a and b (page 202) from the textbook

This assignment requires addressing multiple statistical concepts and problems from HW 5 of STAT 3704, specifically question 1: 4.12 a and b from page 202 of the textbook. The core focus involves understanding p-values, significance levels, hypothesis testing, and interpreting test statistics within the context of a specified null hypothesis about the weight of a McDonald's quarter pounder.

Paper For Above instruction

Statistical hypothesis testing serves as a fundamental tool in inferential statistics, allowing researchers to make decisions about population parameters based on sample data. This paper explores key concepts such as p-values, significance levels (α), and the interpretation of test statistics, contextualized through a practical example involving the weight of a fast-food item.

Understanding P-values and significance level (α)

The p-value quantifies the probability of observing a test statistic as extreme as, or more extreme than, the one calculated from the sample data, assuming the null hypothesis (H₀) is true. It provides a measure of the evidence against H₀, with smaller p-values indicating stronger evidence to reject the null hypothesis. For instance, if the p-value is less than the predetermined significance level, α, then the corresponding null hypothesis is rejected in favor of the alternative hypothesis (Ha).

The significance level, α, is set by the researcher before testing and represents the maximum acceptable probability of making a Type I error—incorrectly rejecting H₀ when it is actually true. Common α values are 0.05, 0.01, and 0.10, depending on the context and desired stringency of the test.

Application: Testing the weight of a McDonald's quarter pounder

In the given scenario, a null hypothesis (H₀) states that the average weight of a McDonald's quarter pounder is 0.25 pounds (H₀: μ = 0.25). The alternative hypothesis (Hₐ) suggests that the weight is less than 0.25 pounds (Hₐ: μ

To determine the largest significance level at which H₀ would not be rejected, we compare the observed z to critical z-values corresponding to various α levels. The p-value associated with z = -2.02 can be found using standard normal distribution tables or statistical software. The p-value is approximately 0.0217, meaning there is about a 2.17% chance of observing such an extreme value if H₀ is true.

If the significance level α is set at 0.05, since 0.0217

Implications of the findings

This analysis highlights the importance of setting an appropriate significance level in hypothesis testing. By understanding the p-value, researchers can determine the strength of evidence required to reject H₀. In practical applications such as quality control or product testing, such statistical evaluations ensure that consumer expectations and safety standards are met. For example, if fast-food outlets aim to ensure their products meet certain weight standards, hypothesis testing can support quality assurance processes.

Broader significance

Understanding p-values and significance levels extends beyond specific examples and is central to scientific research, policy-making, and quality control. The careful interpretation of test statistics allows for informed decision-making, balancing the risks of Type I and Type II errors. Moreover, these statistical tools are applicable across various fields, including healthcare, manufacturing, economics, and social sciences.

Conclusion

In conclusion, the concepts of p-value and significance level are critical in the context of hypothesis testing. The provided example of testing the weight of a McDonald's burger demonstrates how statistical evidence is used to make informed decisions. The p-value helps quantify the strength of evidence against H₀, and the significance level determines the threshold for rejecting H₀. Proper application of these concepts ensures robust and reliable inference in scientific investigations and practical decision-making processes.

References

  • Agresti, A., & Finlay, B. (2009). Statistical methods for the social sciences (4th ed.). Pearson Education.
  • Casella, G., & Berger, R. L. (2002). Statistical inference (2nd ed.). Duxbury.
  • Fisher, R. A. (1925). Statistical methods for research workers. Oliver & Boyd.
  • Moore, D. S., McCabe, G. P., & Craig, B. A. (2012). Introduction to the practice of statistics (8th ed.). W. H. Freeman.
  • Ott, R. L. (2011). An introduction to statistical methods and data analysis (6th ed.). Cengage Learning.
  • Motulsky, H. (2014). Intuitive biostatistics. Oxford University Press.
  • Newcombe, R. G. (1998). Two-sided confidence intervals for the single proportion: Comparison of seven methods. Statistics in Medicine, 17(8), 873-890.
  • Ryan, T. P. (2016). Modern experimental design. Wiley.
  • Siegel, S., & Castellan Jr., N. J. (1988). Nonparametric statistics for the behavioral sciences. McGraw-Hill.
  • Wasserstein, R. L., & Lazar, N. A. (2016). The ASA statement on p-values: Context, process, and purpose. The American Statistician, 70(2), 129-133.

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