Math 221 Statistics For Decision Making Week 4 I-Lab Name

1math 221 Statistics For Decision Makingweek 4 Ilabname

Open a new Excel worksheet. In cell A1 type “success” as the label. Under that, in column A, type 0 through 10 to represent possible successes. In cell B1, type “one fourth” and in cell B2, enter the formula for binomial probability: “=BINOM.DIST(A2,10,0.25,FALSE)”. Copy this formula down through cell B12. Repeat the process in columns C and D for “one half” with formula “=BINOM.DIST(A2,10,0.5,FALSE)” and “three fourths” with formula “=BINOM.DIST(A2,10,0.75,FALSE)” respectively. Then, create scatter plots for each distribution by selecting the data and inserting scatter plots through the Insert menu.

Using the same survey data from Week 2, calculate descriptive statistics—mean and standard deviation—for the variable representing coin flips. Round to three decimal places.

Determine the probabilities for each number of successes (x=0,1,2,...,10) in the binomial experiment with success probability ½, and calculate cumulative probabilities such as P(x ≥1), P(x1), P(x ≤ 4), and 4

Manually compute the mean and standard deviation for the binomial distribution with p=½, n=10, using formulas: Mean = np, Standard deviation = √(npq). Then, repeat these calculations for p=¼, n=10, compare the results in a short paragraph.

Explain why the coin flip data from class fits a binomial distribution based on the properties of binomial experiments. Compare the sample mean and standard deviation obtained from data with those calculated theoretically for p=½, n=10, and discuss their relationship.

Sample Paper For Above instruction

The application of probability distributions, especially the binomial distribution, is fundamental in statistical decision-making processes. In Week 4 of the MATH 221 course, students are tasked with exploring binomial probabilities through Excel, interpreting descriptive statistics, and applying theoretical calculations to real-world data. This comprehensive exercise enhances understanding of how probability models are constructed and validated against observed data.

Using Excel to calculate binomial probabilities involves setting up a spreadsheet with success counts ranging from 0 to 10 and calculating the probability for each count at different success probabilities (¼, ½, ¾). These probabilities illustrate how likely specific outcomes are under different scenarios. The formulas in Excel, such as “=BINOM.DIST(A2,10,0.25,FALSE)", enable quick computation and visualization of the distributions through scatter plots. Visual representations help students comprehend the variability and shape of binomial distributions, which are discrete and inherently probabilistic.

The descriptive statistics—mean and standard deviation—provide vital insights into the data collected from class surveys. For binomial distributions, the mean equals np, and the standard deviation is √(npq). When analyzing the coin flip data, these values can be calculated directly from the data and compared to the theoretical binomial parameters. Consistency between observed and theoretical values confirms the appropriateness of the binomial model.

In calculating probabilities, each individual probability for x successes can be derived using the binomial probability formula. For instance, P(x=0) in the scenario with p=½ and n=10 is calculated as C(10,0)(0.5)^0(0.5)^10. Similarly, probabilities for other values of x can be determined, and cumulative probabilities such as P(x≥1) are obtained through summation. These calculations reveal the distribution's skewness and concentration around the mean.

Manual calculations of the mean and standard deviation reinforce the understanding of the formulas and their application. For p=½, n=10, the mean is 5, and the standard deviation is approximately 1.58, calculated as √(100.50.5). For p=¼, these values shift, leading to a mean of 2.5 and a lower standard deviation, reflecting less variability. Comparing these results illustrates how changes in success probability affect the distribution's shape and spread.

Furthermore, the properties of binomial experiments—fixed number of trials, two possible outcomes, constant probability, and independence—are examined to justify why the coin flip data conforms to a binomial distribution. The observed sample mean and standard deviation are related to theoretical values, and their comparison demonstrates the robustness of the binomial model for such binary outcomes. Minor differences highlight natural sampling variability, but overall, the data aligns well with theoretical expectations.

This comprehensive approach underscores the importance of probability theory in analyzing real-world binary data and decision-making processes. Recognizing how theoretical models align with empirical data provides students with a stronger foundation for statistical reasoning and practical application in diverse fields such as psychology, economics, and health sciences.

References

  • Newbold, P., Carlson, W., & Thorne, B. (2013). Statistics for Business and Economics. Pearson.
  • Moore, D. S., McCabe, G. P., & Craig, B. A. (2017). Introduction to the Practice of Statistics. W.H. Freeman.
  • Upton, G. & Cook, I. (2014). Understanding Statistics. Oxford University Press.
  • Triola, M. F. (2018). Elementary Statistics. Pearson.
  • Wooldridge, J. M. (2013). Introductory Econometrics: A Modern Approach. South-Western Cengage Learning.
  • Everitt, B., & Hothorn, T. (2011). An Introduction to Applied Multivariate Analysis with R. Springer.
  • Johnson, R. A., & Wichern, D. W. (2014). Applied Multivariate Statistical Analysis. Pearson.
  • Agresti, A., & Finlay, B. (2009). Statistical Methods for the Social Sciences. Pearson.
  • Freund, J. E., & Walpole, R. E. (2010). Mathematical Statistics with Applications. Pearson.
  • Ott, R. L., & Longnecker, M. (2010). An Introduction to Statistical Methods and Data Analysis. Cengage Learning.

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