Probability 41 MBB Problem 72 Modified

Probability1probability 41 Mbb Problem 72 Modifi

Analyze and interpret the given statistical problems involving normal, binomial, Poisson, and hypergeometric distributions, focusing on calculating probabilities, percentiles, and assessing the likelihood of specific outcomes within different contexts and datasets.

Use the provided data to solve the problems step-by-step, applying relevant statistical formulas, normal distribution tables or functions, and approximation techniques where necessary. Support your analysis with clear explanations of the methods used, assumptions made, and the implications of the results obtained.

Paper For Above instruction

Analyzing the application of statistical distributions in organizational decision-making fosters a comprehensive understanding of real-world data interpretation. The collection of problems presented involves diverse scenarios, each requiring the application of specific probability distribution concepts such as normal, binomial, Poisson, and hypergeometric distributions. This essay explores these applications in detail, demonstrating how statistical tools inform managerial decisions across different contexts.

The initial set of problems primarily pertains to the normal distribution, which models many biological and social phenomena due to its central limit theorem properties. For example, the average time a manager spends in an employee review is assumed to follow a normal distribution with a known mean of 27.5 minutes and a standard deviation of 2.5 minutes. To determine the percentage of reviews within specific time intervals, z-scores are calculated, converting raw scores into standard normal variates, facilitating the use of the standard normal table or cumulative distribution function (CDF). For instance, the probability that reviews take between 25 and 30 minutes involves calculating z-scores for these bounds and then subtracting their cumulative probabilities. The calculation indicates that approximately 68.27% of reviews fall within this range, aligning with the empirical rule that about 68% of data in a normal distribution lies within one standard deviation from the mean.

Expanding this, the probability that reviews fall between 22.5 and 32.5 minutes is about 95.45%, which reflects the distribution's spread covering roughly two standard deviations from the mean. Extending further, the interval between 20 and 35 minutes captures nearly 99.73% of the reviews, illustrating the typical coverage of data within three standard deviations. Such calculations demonstrate the utility of the standard normal distribution as a tool for probabilistic estimation in time-based managerial assessments.

The problem also discusses the additive property of independent normal distributions. When a manager conducts multiple reviews in sequence, their combined duration follows another normal distribution with a mean equal to the sum of individual means and a consequently adjusted standard deviation, calculated using the square root of the sum of variances. For instance, conducting four reviews results in a mean of 110 minutes and a standard deviation of approximately 3.54 minutes. Applying the z-score for the 120-minute mark (from 10:00 to 12:00) reveals a 97.73% probability that the total time will be less than 120 minutes, which provides a predictive metric for scheduling and resource planning.

The subsequent set of problems involves the estimation of the sample time for reimbursements in a public sector context. The normal distribution with a mean of 36 days and a standard deviation of 5 days is used to determine the maximum processing time to cover 95% of claims, which is calculated as approximately 44.22 days. Another focus is establishing an interval range that captures 95% of reimbursement times, resulting in a range of about 26.2 to 45.8 days. Such percentile estimates help define operational benchmarks and improve transparency for employees awaiting reimbursements. The comparison indicates that setting a maximum threshold may be more practical than a range, especially for communication and policy setting, as it simplifies expectations to a single, clear standard.

In the context of donation analysis, the director's goal is to understand donation patterns. Using the normal distribution with mean $51 and standard deviation $14, the probability that a donation is less than $60 is approximately 73.89%, consistent with the typical behavior where most donations hover around the mean. Conversely, the probability of a donation of $60 or more is about 26.11%. The feasibility of achieving a long-term average donation of at least $80 is analyzed, but the data suggest that such a high average is unlikely under current trends, with only roughly 1.92% of donations meeting that criterion. This insight is critical for strategic planning and fundraising efforts in the organization.

The analysis extends to jury selection, where the likelihood of selecting a jury with a high proportion of Baptists is assessed using binomial distribution approximated by the normal distribution due to sufficiently large sample sizes. If Oklahoma County's population is 40% Baptist, the probability that 22 or more out of 30 randomly selected jurors are Baptists is extremely low (~0.02%). When the available jury pool is 60% Baptist, this probability increases notably to approximately 6.8%. These probabilities inform discussions around potential biases in jury composition and support the use of statistical models to evaluate fairness and randomness in jury selection processes.

Further, the employment service's placement success rates are modeled via the binomial distribution, with a success probability of 0.6 for each individual. Sending seven applicants yields a probability of around 41.99% that at least five will secure jobs, enabling the service to meet its yearly quota. When planning to send eleven applicants, the probability of achieving at least five successful placements exceeds 90%, supporting strategic decisions to scale applicant numbers for effectiveness and certainty.

The last scenario involves the Poisson distribution, suitable for modeling counts of events over time or space with a fixed mean rate. The probability that EcoSystem Inc. secures exactly two contracts in a year, given an average of 0.5 contracts annually, is approximately 7.58%. Extending to two years, the chance of receiving four contracts is calculated, as is the probability of receiving six contracts in three years, which remains low but quantifiable. These estimates assist in risk assessment and resource allocation for project planning with stochastic elements.

Finally, the hypergeometric distribution models the probability of selecting a specific number of Baptists in a sample without replacement. The chance that a sample of 25 TAs contains one-third or fewer Baptists, given a population where one-third are Baptists, is nearly certain (~99.68%), which is logical since the sample proportion matches the population. Increasing the sample to 50 similarly results in an almost certain probability (~100%) of the sample proportion remaining at or below one-third, illustrating the conservative nature of sampling in finite populations. These models support decision-making in election or sampling scenarios where population size and sample size are critical considerations.

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