Math 221 Statistics For Decision Making Week 4 I Lab 543407

4math 221 Statistics For Decision Makingweek 4 Ilabname

Open a new Excel worksheet. In cell A1, type “success”. Under that in column A, type 0 through 10. In cell B1, type “one fourth” and in cell B2, enter the formula “=BINOM.DIST(A2,10,0.25,FALSE)” (use BINOMDIST if using Excel 2007). Copy this formula down to B12. In cell C1, type “one half” and in C2, enter “=BINOM.DIST(A2,10,0.5,FALSE)”, then copy down to C12. In cell D1, type “three fourths” and in D2, enter “=BINOM.DIST(A2,10,0.75,FALSE)”, then copy down to D12. Create scatter plots for these three distributions by selecting the data and inserting scatter plots (dots with no connecting lines).

Using the same class survey results from Week 2, calculate descriptive statistics (mean and standard deviation) for the variable “Coin,” where each of 35 students flipped a coin 10 times. Round answers to three decimal places. Write the mean and standard deviation in the grey area below.

Based on the earlier binomial calculations with success probability ½, list the probability values P(x=0), P(x=1), ..., P(x=10), rounded to three decimal places.

Calculate the probabilities P(x ≥ 1), P(x 1), P(x ≤ 4), P(4

Calculate by hand the mean and standard deviation for the binomial distribution with success probability ½ and n=10, using the formulas: Mean = np; Standard Deviation = √(np(1-p)). Write out your calculations explicitly or explain your process.

Repeat the calculation for a binomial distribution with p=¼ and n=10, and compare the results to those in question 5 with a short paragraph explaining how changes in probability affect the mean and standard deviation.

Calculate by hand the mean and standard deviation for the binomial distribution with p=¾ and n=10, and compare these to the previous cases with a brief explanation based on the formulas. Highlight how the value of p influences the distribution's shape and spread.

Using the four properties of a binomial experiment (fixed number of trials, two mutually exclusive outcomes, constant probability of success, independent trials), explain why the “Coin” variable from the class survey qualifies as a binomial distribution in a short paragraph.

Compare the mean and standard deviation obtained from the class survey data with those calculated by hand in question 5. Discuss how the sample data relate to the theoretical binomial distribution, emphasizing the concepts of expected value and variability.

Paper For Above instruction

The analysis of binomial distributions is fundamental to understanding probability models involving dichotomous outcomes, such as coin flips. In this study, we employ Excel to compute binomial probabilities for various success counts with different success probabilities and visualize these distributions through scatter plots. These graphical representations facilitate intuitive comprehension of how probabilities spread around the expected number of successes, which is directly linked to the mean of the distribution.

Using data collected from a classroom survey, we calculate descriptive statistics, specifically the mean and standard deviation, for the variable “Coin,” capturing the number of successes (heads) out of ten flips per student. These statistics provide a practical perspective on real-world binomial experiments, illustrating how sample data approximates theoretical models.

The probabilities for specific success counts (0, 1, ..., 10) are derived from the binomial probability formula, which accounts for the number of ways to achieve a particular number of successes and the associated likelihood based on the success probability. For p=½, the distribution is symmetric around the mean (which is np=5), with probabilities tapering off towards the extremes of 0 and 10. Calculating cumulative probabilities such as P(x ≥ 1) and P(x ≤ 4) helps in understanding the likelihood of various outcomes, which is useful in hypothesis testing or decision-making processes.

Hand calculations of the mean and standard deviation for different success probabilities highlight the impact of p on the binomial distribution's shape. With p=¼, the distribution skews towards lower success counts, raising the mean to 2 and decreasing variability, reflected in a narrower spread. Conversely, p=¾ skews the distribution toward higher success counts, with the mean increasing to 7.5 and variability increasing correspondingly. These calculations demonstrate the direct proportionality of the mean to np and the standard deviation's dependence on √(np(1-p)).

Furthermore, the criteria defining a binomial experiment—fixed number of trials, independent outcomes, constant probability of success, and mutually exclusive outcomes—are met in the coin-flipping scenario, affirming its classification as a binomial distribution. These properties ensure the statistical validity of applying binomial probability calculations and related analyses to the survey data.

Finally, comparing the sample mean and standard deviation from the class data with the theoretical calculations reinforces the connection between empirical and theoretical probability models. The sample statistics tend to fluctuate around the expected values, illustrating the Law of Large Numbers and the consistency of binomial approximation in practical settings. This comprehensive analysis underscores the importance of understanding binomial distributions in decision-making, risk assessment, and statistical inference.

References

  • Bluman, A. G. (2012). Elementary Statistics: A Step by Step Approach. McGraw-Hill Education.
  • DeGroot, M. H., & Schervish, M. J. (2012). Probability and Statistics (4th ed.). Pearson Education.
  • Moore, D. S., & McCabe, G. P. (2013). Introduction to the Practice of Statistics. W.H. Freeman and Company.
  • Wackerly, D., Mendenhall, W., & Scheaffer, R. (2008). Mathematical Statistics with Applications. Cengage Learning.
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