In A Test Of A Weight Loss Program: Weights Of 40 Sub 892805

In A Test Of A Weight Loss Program Weights Of 40 Subjects Are Recorde

In a test of a weight loss program, weights of 40 subjects are recorded before and after the program. Assume that the before/after weights result in r = 0.876. Is there sufficient evidence to support a claim of a linear correlation between before/after weights? Does the value of r indicate that the program is effective in reducing weight? Why or why not?

Paper For Above instruction

The primary goal of this analysis is to determine whether there is a significant linear correlation between the before and after weights of subjects involved in a weight loss program, and to interpret what the correlation coefficient indicates about the program's effectiveness in reducing weight. The correlation coefficient, r = 0.876, suggests a strong positive linear relationship between the two variables. This means that subjects who initially weighed more tended to weigh more after the program as well, and vice versa.

Statistical Significance of the Correlation

To establish if there is sufficient evidence to support a claim of a linear correlation, we perform a hypothesis test for the correlation coefficient. The null hypothesis (H0) states that there is no correlation between the variables in the population (ρ = 0), while the alternative hypothesis (H1) proposes that there is a correlation (ρ ≠ 0).

Using the sample size of n = 40 subjects and the observed correlation r = 0.876, the test statistic t can be calculated as:

\[ t = r \sqrt{\frac{n-2}{1 - r^2}} \]

Plugging in the numbers:

\[ t = 0.876 \sqrt{\frac{40-2}{1 - 0.876^2}} \]

\[ t = 0.876 \sqrt{\frac{38}{1 - 0.767}} \]

\[ t = 0.876 \sqrt{\frac{38}{0.233}} \]

\[ t = 0.876 \times \sqrt{163.04} \]

\[ t \approx 0.876 \times 12.77 \]

\[ t \approx 11.17 \]

This t-value is compared to the critical t-value from the t-distribution with 38 degrees of freedom (df = n-2). At a typical significance level of α = 0.05, the critical t-value is approximately 2.024. Since 11.17 >> 2.024, we reject the null hypothesis, confirming that there is a statistically significant linear correlation between the before and after weights of the subjects.

Interpretation of the Correlation Coefficient

While the correlation is statistically significant, its magnitude (r = 0.876) indicates a strong correlation, but it does not directly imply causation or effectiveness of the program in weight reduction. Correlation measures the degree of linear association between two variables; high correlation does not necessarily mean the program caused the reduction in weight, especially since the data compares "before" and "after" weights within individuals.

Effectiveness of the Weight Loss Program

To assess whether the program was effective in reducing weight, the average change in weight should be examined. For the program to be deemed effective, on average, subjects should have lost weight after participating in the program. This involves performing a paired t-test to compare the mean weights before and after the program.

The correlation coefficient alone cannot determine effectiveness. Instead, we need insights on the mean difference and its statistical significance. If the mean difference is negative (indicating weight loss) and statistically significant, we can conclude that the program had a substantial effect.

Limitations and Further Analysis

It is important to recognize that a high correlation coefficient like 0.876 indicates consistency in the relationship between variables but does not prove causality or the success of the intervention. Factors such as individual variability, diet adherence, exercise routines, and measurement accuracy may influence the results.

Further, other statistical analyses, such as paired sample t-tests, confidence intervals, and effect size calculations, would provide more comprehensive insights into the program's effectiveness.

Conclusion

Based on the statistical analysis, there is strong evidence supporting a significant linear correlation between the pre- and post-weight measurements of subjects in the weight loss program. However, the correlation alone does not confirm that the program effectively reduces weight; additional analyses focusing on average weight change are necessary. A comprehensive evaluation combining correlation analysis with measures of mean difference and significance testing would better establish the program's efficacy.

References

• Field, A. (2013). Discovering Statistics Using IBM SPSS Statistics. Sage Publications.
• Gravetter, F. J., & Wallnau, L. B. (2016). Statistics for the Behavioral Sciences. Cengage Learning.
• Lehmann, E. L., & Romano, J. P. (2005). Testing Statistical Hypotheses. Springer.
• Moore, D. S., & Notz, W. I. (2013). Statistics: Concepts and Controversies. W. H. Freeman.
• Weiss, N. A. (2012). Introductory Statistics. Pearson.
• Ott, R. L., & Longnecker, M. (2010). An Introduction to Statistical Methods and Data Analysis. Cengage Learning.
• Kim, T., & Kim, D. (2020). Application of correlation analysis in health research. Journal of Medical Data Science, 12(3), 147-155.
• Helsel, D. R., & Hirsch, R. M. (2002). Statistical methods in water resources. United States Geological Survey.
• Tabachnick, B. G., & Fidell, L. S. (2013). Using Multivariate Statistics. Pearson Education.
• Schober, P., Boer, C., & Schwarte, L. A. (2018). Correlation coefficients: appropriate use and interpretation. Anesthesia & Analgesia, 126(5), 1763-1768.