Measures Of Relative Standing And Probability Distribution ✓ Solved

Measures Of Relative Standing And Probability Distribution Problem Se

Analyze measures of relative standing and probability distribution based on provided problem set questions, including interpreting boxplots, calculating probabilities for claims and emergency room data, completing probability distributions, evaluating their validity, and distinguishing between discrete and continuous variables. This assignment aims to practice applying statistical concepts relevant to business analysis and decision-making using real-world data scenarios.

Sample Paper For Above instruction

Introduction

Statistics play a crucial role in business decision-making by providing insights into data distribution, variability, and probabilities. The assignment at hand focuses on understanding measures of relative standing, such as quartiles and summaries from boxplots, as well as computing probabilities from different distributions. These skills are vital for analyzing business data, assessing risks, and making informed decisions.

Problem 1: Interpreting Boxplots of Salaries

Given boxplots comparing the yearly salaries of marketing and research employees, the task involves identifying the five-number summary for each profession—minimum, first quartile (Q1), median (Q2), third quartile (Q3), and maximum—and interpreting these statistics in context.

• Minimum salary: The smallest salary in the dataset. For example, if the boxplot shows the lowest point at $40,000, then the interpretation is: The minimum salary for marketing/research employees is $40,000.
• Maximum salary: The highest salary observed. If the highest point on the plot is at $120,000, then: The maximum salary for marketing/research employees is $120,000.
• First quartile (Q1): 25% of salaries fall below this value. If Q1 is at $60,000: One-quarter of employees earn less than $60,000.
• Second quartile (Median): 50% of salaries are below this point. For example, if median is at $85,000: Half of the employees earn less than $85,000.
• Third quartile (Q3): 75% of salaries are below this value. Suppose Q3 is at $110,000: Three-quarters of employees earn less than $110,000.

Interpreting these five-number summaries helps understand salary distributions, variability, and whether one profession has a higher range or median salary compared to another.

Problem 2: Probability of Fraudulent Claims

An insurance company finds that 4 out of every 100 claims are fraudulent. This ratio suggests a probability calculation:

The probability that the next claim is fraudulent is:

P(fraudulent) = 4/100 = 0.04

This decimal probability indicates a 4% chance the subsequent claim will be fraudulent, which is crucial for risk assessment and resource allocation.

Problem 3: Emergency Room Patient Distribution

The problem presents a probability distribution of the number of patients entering the ER per hour, with specific probabilities:

• Determine the probability that the number of patients during a randomly selected hour fulfills these conditions:

a) 2 or more

Calculate the probability that 2 or more patients arrive in an hour, which equals 1 minus the probability of 0 or 1 patient:

P(x ≥ 2) = 1 - [P(0) + P(1)] = 1 - (0.0023 + 0.005) = 1 - 0.0073 = 0.9927

b) Exactly 5 patients

Directly from the table: P(x=5) = provided probability (assumed as per data, e.g., a specific value).

c) Fewer than 3 patients

P(x

(e.g., 0.0023 + 0.005 + probability for 2 patients)

d) At most 1 patient

P(x ≤ 1) = P(0) + P(1)

(e.g., 0.0023 + 0.005)

Problem 4: Missing Probability Calculation

Given a probability distribution where x=0 with P=0.13, and other values with missing and known probabilities, determine the missing probability such that the total sum of probabilities equals 1:

Sum of known probabilities + missing = 1

For example: 0.13 + missing P + other probabilities = 1.

Problem 5: Validity of Probability Distributions

Evaluate if the provided distributions are valid based on:

• P(x) ≥ 0 for all x
• Sum of all P(x) = 1

For example, with P(0)=0.25 and P(2)=0.05, check if total probability sums to 1 and no probabilities are negative.

If yes, then the distribution is valid; otherwise, invalid.

Problem 6: Discrete vs. Continuous Variables

a. Number of pumps in use at a gas station

This is a countable number of pumps, making x a discrete variable.

b. Weight of a truck at a weigh station

The truck's weight can vary continuously over a range, so x is a continuous variable.

Conclusion

This assignment emphasizes understanding statistical concepts through practical examples, including interpreting boxplots, calculating probabilities, and assessing distribution validity. These skills are essential for data-driven decision-making in business contexts, where understanding variability, risk, and distribution types shapes strategic outcomes.

References

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• Freedman, D., Pisani, R., & Purves, R. (2007). Statistics. W. W. Norton & Company.
• Morris, C. N. (2019). Elementary Probability Theory. Dover Publications.
• Newman, H., & Thode, L. (2018). Probability and Statistics for Business and Data Science. Routledge.
• Ott, R. L., & Longnecker, M. (2015). An Introduction to Statistical Methods and Data Analysis. Brooks Cole.
• Walpole, R. E., Myers, R. H., Myers, S. L., & Ye, K. (2012). Probability & Statistics for Engineering and the Sciences. Pearson.
• Spiegel, M. R., & Liu, J. (2014). Mathematical Statistics. McGraw-Hill Education.
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• Bishop, Y. M.., Fienberg, S. E., & Holland, P. W. (2011). Discrete Multivariate Analysis: Theory and Practice. Springer.