Calculating Percentage Of Snake Lengths Above 16.6 Inches Using Normal Distribution

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In the captivating realm of statistics, the normal distribution, often hailed as the bell curve, reigns supreme as a fundamental concept. Its elegance lies in its ability to model a vast array of natural phenomena, from the heights of individuals to the blood pressure levels within a population. Within this context, we embark on an intriguing exploration of snake lengths, leveraging the power of normal distribution to unravel the percentage of snakes exceeding a specific length threshold. Let's delve into the intricacies of this problem, dissecting the steps involved in arriving at the solution.

The beauty of the normal distribution resides in its symmetrical, bell-shaped form, characterized by two pivotal parameters: the mean (µ), which pinpoints the distribution's center, and the standard deviation (σ), which quantifies the spread or dispersion of the data. In our reptilian scenario, we are presented with a population of snakes whose lengths adhere to a normal distribution, boasting a mean length (µ) of 15 inches and a standard deviation (σ) of 0.8 inches. Our mission is to ascertain the percentage of snakes that stretch beyond the 16.6-inch mark, a task that necessitates a meticulous application of normal distribution principles.

The first step in deciphering this statistical puzzle involves calculating the z-score, a crucial metric that expresses how many standard deviations a particular data point deviates from the mean. The z-score serves as a bridge, transforming our raw data point (16.6 inches) into a standardized value that can be readily interpreted within the context of the standard normal distribution, a normal distribution with a mean of 0 and a standard deviation of 1. The formula for calculating the z-score is elegantly simple:

z = (X - µ) / σ

Where:

  • X represents the data point of interest (16.6 inches in our case).
  • µ denotes the mean of the distribution (15 inches).
  • σ signifies the standard deviation of the distribution (0.8 inches).

Plugging in the values, we obtain:

z = (16.6 - 15) / 0.8 = 2

This z-score of 2 signifies that a snake length of 16.6 inches lies 2 standard deviations above the mean length of 15 inches. This standardized score empowers us to tap into the wealth of information embedded within the standard normal distribution.

Now, armed with the z-score, we turn to the z-table, also known as the standard normal table. This indispensable tool tabulates the cumulative probabilities associated with the standard normal distribution. In essence, the z-table reveals the proportion of data points that fall below a given z-score. Our quest, however, is to determine the percentage of snakes longer than 16.6 inches, which translates to finding the area under the normal curve to the right of our calculated z-score.

Consulting the z-table for a z-score of 2, we unearth a cumulative probability of approximately 0.9772. This value indicates that about 97.72% of snakes possess lengths less than 16.6 inches. To unveil the percentage of snakes exceeding this length, we invoke the fundamental principle that the total area under the normal curve is 1 (or 100%).

Therefore, the percentage of snakes longer than 16.6 inches is calculated as:

1 - 0.9772 = 0.0228

This result translates to approximately 2.28%. Considering the options provided, the closest answer is B. 2.5 %.

The normal distribution, also known as the Gaussian distribution, is a cornerstone of statistics, characterized by its symmetrical bell-shaped curve. Its ubiquity stems from its ability to model a wide spectrum of natural phenomena, from physical measurements to psychological attributes. Understanding the nuances of normal distribution is paramount for anyone venturing into the realms of data analysis and statistical inference.

Key Properties of Normal Distribution:

  • Symmetry: The normal curve is perfectly symmetrical around its mean, implying that the left and right halves mirror each other.
  • Unimodality: The normal distribution possesses a single peak, which coincides with the mean, median, and mode of the data.
  • Mean, Median, and Mode: In a normal distribution, the mean, median, and mode are all equal, a consequence of its symmetrical nature.
  • Empirical Rule: This rule provides valuable insights into the distribution of data within a normal curve:
    • Approximately 68% of the data falls within one standard deviation of the mean.
    • Around 95% of the data lies within two standard deviations of the mean.
    • Virtually all (99.7%) of the data resides within three standard deviations of the mean.
  • Standard Normal Distribution: A special case of the normal distribution, the standard normal distribution, boasts a mean of 0 and a standard deviation of 1. Its significance lies in its role as a reference distribution for standardizing other normal distributions.

Applications of Normal Distribution:

The versatility of the normal distribution extends to numerous domains:

  • Natural Sciences: Modeling physical measurements, such as heights, weights, and blood pressure levels.
  • Social Sciences: Analyzing psychological traits, such as intelligence and personality traits.
  • Finance: Forecasting stock prices and assessing investment risks.
  • Engineering: Controlling quality in manufacturing processes.
  • Healthcare: Evaluating the effectiveness of medical treatments.

Common Misconceptions about Normal Distribution:

  • Normality is ubiquitous: While normal distribution is prevalent, not all data conforms to its pattern. It's crucial to assess the data's distribution before assuming normality.
  • Normality implies perfection: Normality merely indicates a specific data pattern, not necessarily ideal conditions. Some scenarios might necessitate non-normal distributions.
  • Mean is the sole descriptor: While the mean is vital, the standard deviation provides crucial information about data spread. A sole focus on the mean can be misleading.

Z-scores, also known as standard scores, serve as invaluable tools for standardizing data within a normal distribution. They quantify the distance between a data point and the mean in terms of standard deviations. A positive z-score indicates that the data point lies above the mean, while a negative z-score signifies a position below the mean. A z-score of 0 implies that the data point coincides with the mean.

Interpreting Z-scores:

  • Magnitude Matters: The absolute value of the z-score reveals the distance from the mean. Larger absolute values signify greater deviations from the mean.
  • Direction Matters: The sign of the z-score indicates the data point's position relative to the mean. Positive values denote above-average scores, while negative values denote below-average scores.
  • Context is Key: The interpretation of z-scores is context-dependent. A z-score of 2 might be significant in one scenario but mundane in another.

Applications of Z-scores:

  • Data Comparison: Z-scores facilitate comparisons between data points from different distributions. Standardizing the data through z-scores allows for meaningful comparisons.
  • Outlier Detection: Z-scores aid in identifying outliers, data points that deviate significantly from the norm. Data points with z-scores exceeding a predefined threshold (e.g., 2 or 3) might warrant further scrutiny.
  • Probability Calculation: Z-scores pave the way for calculating probabilities associated with normal distributions. By consulting the z-table, we can determine the proportion of data points falling above or below a specific value.

In summary, we have successfully navigated the statistical landscape of snake lengths, employing the power of normal distribution and z-scores to ascertain the percentage of snakes exceeding 16.6 inches. This journey has not only furnished us with a concrete answer (approximately 2.5%) but also illuminated the broader significance of normal distribution in modeling real-world phenomena. By grasping the nuances of normal distribution, z-scores, and z-tables, we equip ourselves with invaluable tools for data analysis and decision-making in a multitude of contexts.

The normal distribution, with its elegant bell curve and well-defined properties, serves as a cornerstone of statistical analysis. Its ability to model a wide range of natural and social phenomena makes it an indispensable tool for researchers, analysts, and anyone seeking to extract insights from data. The z-score, as a standardized measure of deviation from the mean, further empowers us to compare data points across different distributions and calculate probabilities with precision. As we continue to explore the world of statistics, the principles of normal distribution and z-scores will undoubtedly serve as guiding lights, illuminating our path toward deeper understanding and informed decision-making. In the realm of data, knowledge is power, and the normal distribution is a powerful ally indeed.

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