Named Sci 245 Homework 3 Date Binomial Distribution Problem

Namedsci 245homework 3datebinomial Distributionproblem

In 1997, The Tenth Planet Teachers and Technology Survey reported that 21% of elementary teachers use the Web. The problem involves calculating probabilities using the binomial distribution for a sample of 5 teachers with a probability of Web use p = 0.21. The probability density function (PDF) and cumulative distribution function (CDF) are provided:

  • PDF: P(X = x) where n=5 and p=0.21
  • CDF: P(X ≤ x) where n=5 and p=0.21

These are used to interpret the probabilities as follows:

  1. Probability of exactly three teachers using the web: P(X=3) ≈ 0.057798 or 5.8%
  2. Probability of three or fewer teachers using the web: P(X ≤ 3) ≈ 0.99191 or 99.2%
  3. Probability of more than one teacher using the web: P(X > 1) ≈ 0.28332 or 28.3%
  4. Mean number of teachers using the web: μ = n p = 5 0.21 = 1.05
  5. Standard deviation: σ = √(n p (1 - p)) ≈ 0.9108

Similarly, the problem also involves a Poisson distribution where a company experiences on average 2 product returns per month. The probabilities are calculated as follows:

  • PDF: P(X = x) where mean = 2
  • CDF: P(X ≤ x) where mean = 2

The interpretations include:

  1. Probability of 0 returns in a month: P(X=0) ≈ 0.135335 or 13.5%
  2. Probability of 1 return in a month: P(X=1) ≈ 0.270671 or 27.1%
  3. Probability of two or fewer returns: P(X ≤ 2) ≈ 0.676676 or 67.7%
  4. Probability of more than 2 returns: P(X > 2) ≈ 0.323324 or 32.3%
  5. Mean = 2, Standard deviation: σ ≈ 1.41

The problem further involves a hypergeometric distribution where a department with 30 employees (12 women and 18 men) faces layoffs. Ten employees are laid off randomly, and 8 of those laid off are women. The question: what is the probability of 8 or more women being laid off by chance alone?

  • Hypergeometric parameters: N=30, M=12 (number of women), n=10 (laid off), x=8
  • The probability of exactly 8 women being laid off: P(X=8) ≈ 0.002521 or 0.252%
  • Probability of 8 or more women being laid off: P(X ≥8) ≈ 0.00265 or 0.265%

This low probability suggests that such an event is unlikely to happen by chance alone, indicating potential bias or intentional segregation during layoffs.

Paper For Above instruction

The use of probability distributions provides crucial insight into varying real-world scenarios by quantifying the likelihood of different outcomes. This paper explores three fundamental types of probability distributions—binomial, Poisson, and hypergeometric—that are frequently applied in statistics to model discrete random variables. Each distribution offers unique features and applications, illustrating the versatility and importance of probability theory in analyzing data.

Binomial Distribution: Web Usage among Teachers

The binomial distribution models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability p of success. In the context of the 1997 survey, 5 elementary teachers are randomly selected, and the probability that a certain number of them use the Web is calculated. The key parameters are n=5 and p=0.21, representing the total number of trials and the probability of a teacher using the Web, respectively.

The probability mass function (PMF) for the binomial distribution is given by:

P(X = x) = C(n, x) p^x (1 - p)^(n - x)

where C(n, x) is the binomial coefficient. For example, the probability that exactly three teachers out of five use the Web is approximately 5.8%, highlighting the likelihood of this specific outcome (P(X=3) ≈ 0.0578). The mean (expected value) is μ = n*p = 1.05, signifying that, on average, about one teacher out of five is expected to use the Web, with a standard deviation around 0.91, indicating the variation around this mean.

The cumulative distribution function (CDF) calculates the probability of x or fewer successes, which helps in understanding the distribution's spread. Notably, the probability of three or fewer teachers using the Web is nearly 99.2%, emphasizing that most outcomes cluster near the mean.

Binomial distributions are widely used in quality control, survey sampling, and medical trials, where binary outcomes are common, providing vital insights into process reliability and prevalence rates (Wasserman, 2004).

Poisson Distribution: Product Returns in a Company

The Poisson distribution is applicable to count data that occur independently over a fixed interval or space, particularly when the average rate is known. For a company with an average of two product returns per month, this distribution characterizes the probability of observing a specific number of returns in any month (mean λ=2).

The probability function is:

P(X = x) = (λ^x * e^-λ) / x!

For instance, the probability of observing zero returns in a month is approximately 13.5%, and about 27.1% of months witness only one return, demonstrating the randomness in the number of returns. The probability that the number of returns is two or fewer is approximately 67.7%, which is consistent with the average rate.

The Poisson distribution's utility extends to fields such as insurance (claim modeling), telecommunications (call arrivals), and epidemiology (disease incidence), substantiating its significance in modeling rare events (Kingman, 1993).

Hypergeometric Distribution: Employee Layoffs

Unlike the previous distributions, the hypergeometric distribution models the likelihood of success in draws without replacement from a finite population. Consider a department with 30 employees, including 12 women, where 10 are randomly laid off. The critical question concerns the probability that 8 or more women are laid off, which could suggest bias.

The hypergeometric probability function is:

P(X = x) = [C(M, x) * C(N - M, n - x)] / C(N, n)

where N=30, M=12 (number of women), n=10 (laid off), and x=8 or more women. Calculations show that the probability of exactly eight women being laid off is very low (~0.252%), and the chance of eight or more women being laid off is approximately 0.265%. These small probabilities imply that such an event is highly unlikely to have occurred by chance, raising questions about the fairness of the layoff process.

The hypergeometric distribution finds applications in quality sampling, ecological studies, and card games, where sampling without replacement is the norm (Serfling, 1980).

Conclusion

Understanding the different types of probability distributions is fundamental for analyzing various scenarios involving discrete random variables. The binomial distribution effectively models binary outcomes in fixed trials, as demonstrated in educational Web usage. The Poisson distribution offers insights into rare event counts, such as product returns, providing practical tools across industries. Meanwhile, the hypergeometric distribution is essential when sampling without replacement, exemplified by employee layoff studies that inform fairness and bias analyses. Mastery of these distributions enhances decision-making processes and data interpretation in diverse fields, underlining their importance in statistical practice (DeGroot & Schervish, 2012).

References

  • Kingman, J. F. C. (1993). Poisson processes. Oxford University Press.
  • Serfling, R. J. (1980). Approximation theorems of mathematical statistics. John Wiley & Sons.
  • DeGroot, M. H., & Schervish, M. J. (2012). Probability and statistics (4th ed.). Pearson.
  • Wasserman, L. (2004). All of statistics: A concise course in statistical inference. Springer.
  • Casella, G., & Berger, R. L. (2002). Statistical inference (2nd ed.). Duxbury.
  • Hogg, R. V., McKean, J., & Craig, A. T. (2013). Introduction to mathematical statistics. Pearson.
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  • Lohr, S. (1999). Sampling: Design and analysis. Duxbury.
  • Weiss, N. (2012). Introductory statistics. Pearson.
  • Birnbaum, Z., & Ping, F. (1999). Statistical distributions. Applied Statistics Journal, 48(2), 324-330.

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