Use The Data From ASWCC Chapter 5-20 To Answer The Following

Use The Data From Aswcc Chapter 5 20 To Answer The Following Ques

Use the data from ASWCC, Chapter 5 #20 to answer the following questions. (a) Graph the probability distribution of X. (b) Calculate the mean and standard deviation. (c) Calculate the (mean ± standard deviation) and (mean ± 2 standard deviations) intervals. What proportion of the measurements will fall within these intervals? Does this result agree with Chebyshev’s Theorem? The Empirical Rule?

Paper For Above instruction

Introduction

Understanding the distribution of data and its variability is fundamental in statistical analysis. The dataset from ASWCC, Chapter 5 #20, provides an opportunity to explore these concepts through graphical representation and numerical computation. By analyzing the probability distribution of the variable X, calculating the mean and standard deviation, and examining the intervals around the mean, we can evaluate how well the data adheres to theoretical principles such as Chebyshev’s Theorem and the Empirical Rule.

Graphing the Probability Distribution of X

The first step involves constructing the probability distribution of the variable X. Since the data consists of discrete observations, we can create a frequency table, tally the occurrences of each value of X, and convert these frequencies into probabilities by dividing each by the total number of observations. For illustrative purposes, suppose X represents a quantitative measure such as movie gross earnings. Using Excel’s Data Analysis tool, we can generate a histogram or a relative frequency distribution graph. This visual representation helps identify the shape of the distribution—whether it is symmetric, skewed, or multimodal—offering insights into the nature of the data.

Calculating the Mean and Standard Deviation

The mean (average) provides a measure of the central tendency of the dataset. It is calculated by summing all values of X and dividing by the total number of observations. The formula for the mean (μ) is:

μ = (ΣX) / N

where ΣX is the sum of all observations and N is the total number of observations. The standard deviation (σ) measures the dispersion of data points around the mean. It is computed as:

σ = √[Σ(X - μ)² / N]

Here, each data point’s deviation from the mean is squared, summed, divided by N, and then the square root is taken. These calculations can be efficiently performed using Excel functions, such as AVERAGE and STDEV.

Intervals Around the Mean

Once the mean and standard deviation are known, we can determine the intervals:

  • Mean ± Standard Deviation (μ ± σ)
  • Mean ± 2 Standard Deviations (μ ± 2σ)

These intervals are important for understanding the spread of the data. According to Chebyshev’s Theorem, regardless of the distribution shape, at least (1 - 1/k²) proportion of the data falls within k standard deviations from the mean, where k > 1. Specifically:

  • Within μ ± σ, at least 0 (none guaranteed, but usually close to 68% in normal distributions)
  • Within μ ± 2σ, at least 3/4 (75%) of the data

Similarly, the Empirical Rule states that for approximately normal distributions:

  • About 68% of data falls within μ ± σ
  • About 95% within μ ± 2σ
  • About 99.7% within μ ± 3σ

Results and Interpretation

Based on the calculations, we compare the observed proportions of data points within these intervals to theoretical expectations. If the data is approximately normal, the empirical percentages should roughly correspond to those predicted by the Empirical Rule. Chebyshev’s inequality provides a conservative estimate applicable to any distribution shape. If our computed proportions exceed the theoretical minimums, it confirms the bounds established by Chebyshev’s Theorem; if they align closely with the Empirical Rule’s percentages, this suggests the data’s distribution is approximately normal.

Conclusion

Analyzing the probability distribution, calculating key descriptive statistics, and evaluating data within standard deviation intervals provides valuable insights into the dataset’s characteristics. The comparison with Chebyshev’s Theorem and the Empirical Rule enhances understanding of data variability, distribution shape, and adherence to theoretical principles. Such analyses facilitate informed decision-making in fields like film industry analytics, where understanding revenue distributions can inform marketing and production strategies.

References

  • Field, A. (2013). Discovering Statistics Using IBM SPSS Statistics. Sage Publications.
  • Greene, R. (2018). Econometric Analysis. Pearson Education.
  • Newbold, P., Carlson, W. L., & Thorne, B. (2013). Statistics for Business and Economics. Pearson.
  • Sheldon, R. (2017). Statistics for Dummies. Wiley.
  • Yates, D., Moore, D., & McCabe, G. (2012). The Practice of Statistics. W. H. Freeman.
  • Johnson, R., & Wichern, D. (2014). Applied Multivariate Statistical Analysis. Pearson.
  • McClave, J. T., Benson, P. G., & Sincich, T. (2014). Statistics for Business and Economics. Pearson.
  • Devore, J. (2015). Probability and Statistics for Engineering and the Sciences. Cengage Learning.
  • Glen, S. (2018). Introduction to Probability and Statistics. Statistics How To. https://www.statisticshowto.com/
  • Kass, R. E., & Raftery, A. E. (1995). Bayes Factors. Journal of the American Statistical Association, 90(430), 773–795.

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