Inte 296 Assignment 1 Due Date February 3, 2015

Inte 296 Assignment 1assignment 1due Date February 3rd, 2015 Corr

Evaluate various statistical and probability problems based on given data, including measures of central tendency, variability, positional statistics, probability calculations for different scenarios, and concepts of independence and dependence in probability.

Paper For Above instruction

Introduction

The realm of statistics and probability forms a fundamental aspect of analytical decision-making across multiple fields. This paper addresses a series of problems that involve calculating measures of central tendency and variability, understanding positional statistics, applying probability theory to real-world scenarios, and understanding the concepts of independent and dependent events in probability. Each problem case provides a context-rich scenario that exemplifies the practical utility of statistical calculations and probability concepts, illustrating their importance in operational, health, sports, investigative, technical, and everyday decision-making processes.

Question 1: Statistical Analysis of DAST Scores

Globally, assessing employee mental resilience during organizational change involves evaluating their psychological state using standardized tests. The Depression, Anxiety, and Stress Test (DAST) provides a numerical measurement of anxiety levels, and analyzing its scores offers insights into the overall psychological preparedness of employees in transitioning to new product lines.

Given a sample of 20 employees' DAST scores, the analysis begins by calculating measures of central tendency: the mean, median, and mode. The mean provides the average score, indicating the typical anxiety level among employees. The median helps identify the middle score, providing a positional sense less influenced by outliers. The mode reveals the most frequently occurring score, highlighting any common anxiety levels.

With the scores at hand, measures of variability are computed, including the range and standard deviation. The range offers a simple measure of how spread out the scores are, computed as the difference between the maximum and minimum scores. Standard deviation quantifies the average deviation of scores from the mean, capturing the overall dispersion and variability within employee responses.

Measures of position, such as the 15th percentile (P15), the 66th percentile (P66), and the five-number summary (minimum, Q1, median, Q3, maximum), further contextualize the distribution. Percentiles indicate the score thresholds below which a given percentage of scores falls, illuminating the distribution's shape and identifying thresholds for concern or resilience.

Question 2: Golf Score Analysis and Improvement Strategies

In assessing athletic performance, especially golf, gathering and analyzing data over a season helps players determine progress. Christy's recorded scores over the season serve as valuable data points for statistical analysis. Calculating measures of central tendency—mean, median, and mode—reveals typical scoring performance and highlights the most common scores, which together reflect her general level of performance and consistency.

Variability and positional measures such as standard deviation, interquartile range (IQR), the 45th percentile (P45), and the 90th percentile (P90) are essential for understanding score consistency and distribution shape. A lower standard deviation or IQR suggests more consistent performance, while percentile scores indicate how her performance compares across the season, offering insight into improvements over time.

To assess her progress, Christy should consider specific strategies. First, plotting her scores over time can visually reveal trends of improvement or stagnation. Second, calculating the difference between her initial and recent median scores can quantitatively measure progress. Third, incorporating additional variables such as course difficulty, weather conditions, and types of clubs used can help account for external factors influencing scores. Regularly updating her statistical analysis, perhaps monthly, can detect subtle changes and validate her training effectiveness.

Question 3: Probability of Pinkerton Agents’ Attributes and Operations

The Pinkerton Detective Agency's data provides a foundation for computing probabilities of various attributes of its agents. With the known distribution of agents across departments, specializations, and geographic zones, theoretical probabilities are calculated as the ratio of the number of agents fitting specific criteria over the total number of agents.

For instance, the probability that a randomly chosen agent is a Security Console Operator is obtained by dividing the number of agents in that role by the total number of agents across all regions and divisions. Similarly, probabilities for other criteria such as department, geographic location, or specialization are derived by dividing the relevant counts by the overall total. Conditional probabilities—such as the likelihood of an agent working in Puerto Rico given they are a Document/Information Security Specialist—are calculated using the ratio of the joint probability over the probability of the known condition.

Furthermore, the question of independence between attributes, such as being a Technical Support Officer and working in Puerto Rico, involves checking whether the joint probability equals the product of marginal probabilities. If they are not equal, the two attributes are dependent, indicating a relationship or correlation between such characteristics among agents.

Question 4: Battery Fault and Reliability Analysis

Battery reliability is a critical factor in electronic devices. In this scenario, selecting two batteries from a box containing a mixture of functional and faulty batteries involves computing probabilities associated with the quality of the batteries selected.

The probability that the remote control operates correctly, given that it requires two good batteries, depends on selecting two functional batteries from the total pool. This involves combinatorial calculations: the total number of ways to select two batteries and the favorable subsets where both are functional. The probability adapts as batteries are removed after each selection, considering whether the batteries are faulty or functional.

Similarly, if the remote works with the first two batteries, the probability that the next two selections will also be all good involves updating remaining counts and recalculating based on reduced totals. Such analysis informs maintenance and quality control decisions regarding battery manufacturing and inventory management.

Question 5: Analyzing Carol’s Transportation Choices and Timeliness

Understanding Carol's commuting preferences and punctuality involves conditional and total probability calculations. Her choice distribution reflects her preferences: the metro (35%), bus (55%), and driving (10%). The reliability of each method varies, with the probability of being on-time or late being different for each mode.

The overall probability that Carol is late to class incorporates the weighted chance of being late given each mode, combined with her mode preferences. Calculating this involves summing the products of the probability of choosing each mode and the probability of being late given that mode.

Further, the probability that she used the metro and was on time involves multiplying her choice probability by the probability of punctuality for that mode. The same logic applies when analyzing the driving mode.

Conditional probabilities such as the likelihood she took the bus given lateness, or being on-time given she did not drive, are computed using Bayes' theorem and the law of total probability, providing insights into her punctuality pattern and helping inform potential improvements in her commutes.

Question 6: Demonstrating Conditional Probability through a Custom Problem

To elucidate the concept of conditional probability, consider the following scenario: In a small town, 10% of households own a pet cat, and 15% own a dog. Among households that own cats, 50% also own dogs, whereas only 20% of households without cats own dogs. Determine the probability that a randomly selected household owns a dog given that it owns a cat, and contrast this with the probability that a household owns a dog given it does not own a cat.

Solving this problem involves calculating joint probabilities and applying Bayes’ theorem. The problem reveals that owning a cat influences the likelihood of owning a dog, an example of dependent events. Alternatively, if owning a dog were independent of owning a cat, the probability of owning a dog would be the same regardless of pet ownership status.

This example demonstrates when probabilities are dependent—conditional upon pet ownership—versus independent. It helps clarify how the presence or absence of one attribute influences the probability of another, an essential concept in statistical inference and decision-making.

In summary, understanding and calculating conditional and independent probabilities help elucidate real-world relationships between variables, supporting better data-driven decisions in diverse contexts.

Conclusion

Through detailed analysis of the provided scenarios, this paper underscores the vital role of statistical measures and probability theory in interpreting data and making informed decisions. Whether analyzing employee stress levels, sports performance, detective agency attributes, technical reliability, or everyday choices, these tools enable a deeper understanding of underlying patterns and relationships. Mastery of these concepts empowers analysts and decision-makers alike to interpret complex data accurately and to develop strategies grounded in quantitative evidence.

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