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For these project assignments throughout the course you will need to reference the data in the ROI Excel spreadsheet. Download it here. Using the ROI data set: What are the characteristics of a binomial experiment? Explain how the following scenario fits EACH of the requirements for a binomial experiment. Scenario: We select 7 colleges from a major and then record whether they are of ‘School Type’ ‘Private’ or not.

For each of the 2 majors determine if the ‘Annual % ROI’ appears to be normally distributed. Use a histogram and the measures of central tendency (mean and median) to justify your results. In a highlighted box, explain how having or not having a normal distribution for each of the majors may affect your ability to compare the data of the two majors.

Paper For Above instruction

The analysis of data in business and educational research often involves understanding the distribution and characteristics of various data points. This paper explores the properties of binomial experiments, applies these concepts to a specific scenario involving college data, and examines the distribution of Return on Investment (ROI) percentages across different majors. By doing so, it highlights the significance of distribution shapes and their implications on data comparison and decision-making.

Understanding the Characteristics of a Binomial Experiment

A binomial experiment is a statistical experiment designed around independent trials with two possible outcomes, commonly termed success or failure. It satisfies several specific characteristics:

  • Fixed number of trials: The experiment involves a predetermined number of independent repetitions (trials).
  • Two possible outcomes: Each trial results in either success or failure.
  • Constant probability of success: The probability of success remains the same in all trials.
  • Independent trials: The outcome of one trial does not influence another.

Applying these characteristics to the given scenario of selecting colleges, if we define success as “a college being of ‘Private’ school type,” then selecting each college is a Bernoulli trial with two outcomes: Private or not. Assuming random sampling, the probability of selecting a private college remains constant across each selection, and each selection is independent, fulfilling the criteria for a binomial experiment.

Evaluation of the Scenario in the Context of Binomial Experiment Characteristics

In the described scenario—selecting 7 colleges and recording whether each is of ‘School Type’ Private or not—the experiment fits multiple characteristics of a binomial experiment:

  1. Fixed number of trials: The number of colleges selected (7) is fixed, satisfying this criterion.
  2. Two outcomes per trial: Each college is categorized as either private or not private, confirming the binary nature.
  3. Constant probability: Assuming the selection process is random, the probability that any selected college is private remains consistent across trials.
  4. Independence of trials: If colleges are chosen randomly without replacement, the independence assumption might be slightly compromised unless sampling is done with or adjusted for the finite population. However, for large populations, this approximation is acceptable.

Therefore, this specific scenario closely adheres to the defining properties of a binomial experiment, enabling the use of binomial probability models for analyzing the likelihood of a certain number of private colleges among the selected seven.

Distribution Analysis of ‘Annual % ROI’ for Two Majors

Understanding whether the ROI data for each major follows a normal distribution involves visual and statistical analysis. Histograms provide a visual impression of the data distribution, revealing skewness, kurtosis, or symmetry. Measures of central tendency—mean and median—offer quantitative insights into the data’s symmetry and distribution shape.

For each major, histograms display the frequency of ROI percentages, allowing comparison against the bell-shaped curve characteristic of normality. A roughly symmetric histogram centered around a single peak indicates an approximately normal distribution. Conversely, skewed or multi-modal histograms suggest deviations from normality.

The mean and median further clarify distribution shape. When the data are normally distributed, the mean and median are close in value. Significant differences suggest skewness or other departures from normality. For example, if the mean exceeds the median, the data might be right-skewed; if the median exceeds the mean, it could be left-skewed.

Implications of Distribution Shape on Data Comparison

The shape of the ROI distribution impacts the methods and accuracy of comparing the two majors. Normal distribution assumptions underpin many parametric statistical tests, such as t-tests, which compare means. When the data are approximately normal, these tests produce reliable inferences about differences between groups.

If ROI data for a major are not normally distributed, applying parametric tests might lead to misleading conclusions due to violations of underlying assumptions. Alternative non-parametric methods, such as the Mann-Whitney U test, may then be more appropriate for comparing the majors. Moreover, non-normality affects the calculation and interpretation of confidence intervals and effect sizes.

In practical decision-making, understanding the distribution allows researchers to select suitable statistical tools, ensuring accurate analysis and reliable conclusions. Recognizing deviations from normality also guides data transformation or alternative analysis approaches, like bootstrapping or robust statistical methods.

Conclusion

In conclusion, the binomial experiment's characteristics can be effectively demonstrated through a college selection scenario involving ‘School Type’ categorization. The analysis of ROI data distribution across two majors emphasizes the importance of histogram visualization and measures such as mean and median. The distribution shape critically influences the choice of statistical tests and the validity of comparing data sets. Recognizing and accounting for distribution characteristics is essential for sound statistical analysis and meaningful interpretations in research contexts.

References

  1. Field, A. (2013). Discovering statistics using IBM SPSS statistics. Sage.
  2. Gravetter, F. J., & Wallnau, L. B. (2016). Statistics for the behavioral sciences. Cengage Learning.
  3. Myers, J. L., Well, A. D., & Lorch, R. F. (2010). Research Design and Statistical Analysis. Routledge.
  4. Moore, D. S., McCabe, G. P., & Craig, B. A. (2012). Introduction to the Practice of Statistics. W. H. Freeman.
  5. Zimmerman, D. W. (1997). A note on the normality assumption in multivariate analysis. Journal of Educational and Behavioral Statistics, 22(3), 275-280.
  6. Wilcox, R. R. (2012). Introduction to Robust Estimation and Hypothesis Testing. Academic Press.
  7. Sheskin, D. J. (2010). Handbook of Parametric and Nonparametric Statistical Procedures. Chapman & Hall/CRC.
  8. Johnson, R. A., & Wichern, D. W. (2014). Applied Multivariate Statistical Analysis. Pearson.
  9. Kotz, S., & Nadarajah, S. (2004). Extreme Value Distributions: Theory and Applications. Imperial College Press.
  10. Cohen, J. (1988). Statistical Power Analysis for the Behavioral Sciences. Routledge.

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