Z-Score Calculation For Appliance Lifespan Stopped Working At 64 Months

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In the realm of statistics, understanding data distribution is paramount. One crucial concept in this domain is the z-score, also known as the standard score. The z-score quantifies how many standard deviations a particular data point deviates from the mean of its dataset. This metric is invaluable for comparing data points from different datasets and for assessing the probability of occurrence of specific values within a distribution. In this article, we will delve into the application of z-scores in the context of appliance lifespans. Specifically, we will address the question: "What is the z-score of an appliance that stopped working at 64 months, given that the set of lifespans of an appliance is normally distributed with a mean (μ) of 48 months and a standard deviation (σ) of 8 months?"

Before diving into the calculation, let's solidify our understanding of z-scores. A z-score is a dimensionless quantity that represents the distance between a data point and the mean of the dataset, measured in units of standard deviations. A positive z-score indicates that the data point is above the mean, while a negative z-score signifies that it is below the mean. A z-score of zero implies that the data point is equal to the mean.

The formula for calculating the z-score is as follows:

z = (x - μ) / σ

Where:

  • z is the z-score
  • x is the data point
  • μ is the mean of the dataset
  • σ is the standard deviation of the dataset

In our scenario, we are given that the lifespans of appliances follow a normal distribution. This is a crucial piece of information, as normal distributions are ubiquitous in statistics and exhibit well-defined properties. A normal distribution is characterized by its bell-shaped curve, with the mean representing the center of the distribution and the standard deviation dictating the spread of the curve. The empirical rule, also known as the 68-95-99.7 rule, states that for a normal distribution:

  • Approximately 68% of the data falls within one standard deviation of the mean.
  • Approximately 95% of the data falls within two standard deviations of the mean.
  • Approximately 99.7% of the data falls within three standard deviations of the mean.

This rule provides a framework for interpreting z-scores. For instance, a z-score of 2 indicates that the data point is approximately at the 97.5th percentile of the distribution (mean + 2 standard deviation), meaning that only 2.5% of the data points are expected to be higher.

Now, let's apply the z-score formula to our specific problem. We have:

  • x = 64 months (the lifespan of the appliance)
  • μ = 48 months (the mean lifespan)
  • σ = 8 months (the standard deviation)

Plugging these values into the formula, we get:

z = (64 - 48) / 8
z = 16 / 8
z = 2

Therefore, the z-score of an appliance that stopped working at 64 months is 2.

A z-score of 2 signifies that the lifespan of the appliance (64 months) is two standard deviations above the mean lifespan (48 months). This indicates that the appliance lasted longer than most appliances in the set. Based on the empirical rule, we can infer that this appliance's lifespan falls within the top 2.5% of lifespans in the distribution. This calculation and interpretation of the z-score are crucial for understanding the appliance's performance relative to the average lifespan and the variability within the data set.

The z-score is a powerful tool with a wide array of applications beyond the realm of appliance lifespans. In statistics and data analysis, z-scores are used for:

  • Identifying outliers: Data points with z-scores exceeding a certain threshold (e.g., 2 or 3) are often considered outliers, which may warrant further investigation.
  • Comparing data from different distributions: Z-scores allow us to compare data points from different datasets with varying means and standard deviations, as they standardize the data.
  • Hypothesis testing: Z-scores are used in hypothesis testing to determine the statistical significance of results.
  • Quality control: In manufacturing and other industries, z-scores are used to monitor the consistency of processes and identify deviations from expected values.
  • Risk assessment: In finance, z-scores can be used to assess the creditworthiness of individuals or companies.

For instance, in a manufacturing setting, if a machine is producing parts with a dimension that has a z-score of -2.5, it suggests that the parts are significantly smaller than the average, signaling a potential issue with the machine's calibration. Similarly, in financial analysis, a company with a low z-score may be flagged as a high-risk investment due to its financial metrics being far from the industry average.

In conclusion, the z-score of an appliance that stopped working at 64 months, given a mean lifespan of 48 months and a standard deviation of 8 months, is 2. This signifies that the appliance's lifespan is two standard deviations above the average, placing it among the longer-lasting appliances in the dataset. The z-score is a versatile statistical tool that provides valuable insights into data distribution and facilitates comparisons across different datasets. Its applications span various fields, including quality control, finance, and scientific research. Understanding z-scores is essential for anyone working with data, as it enables informed decision-making and a deeper comprehension of the underlying patterns and trends.

By grasping the concept of z-scores and their applications, analysts and decision-makers can derive meaningful insights from data, leading to more effective strategies and interventions in diverse domains. Whether assessing the performance of appliances, evaluating financial risk, or monitoring manufacturing processes, z-scores offer a standardized metric for comparing and interpreting data points relative to their distribution.