A Variable Is Normally Distributed With A Mean Of 12 And A Standard Deviation Of 2. A. Find The Quartiles Of The Variable. B. Determine And Interpret The 90th Percentile. C. Determine The Value That 65% Of All Possible Values Of The Variable Exceed. D. Discuss The Concepts Used.

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In statistics, the normal distribution, often called the Gaussian distribution, is a continuous probability distribution that is symmetrical around the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. In simpler terms, it's a way to understand how data is spread out. Many natural phenomena, like height, weight, and test scores, follow a normal distribution. Understanding normal distribution is critical in many fields, from science to finance, because it allows us to make predictions and interpretations about data. This article delves into the practical applications of normal distribution by exploring quartiles, percentiles, and values exceeding specific percentages within a given dataset. We'll work through a series of problems related to a normally distributed variable with a mean of 12 and a standard deviation of 2, providing a step-by-step guide to understanding these concepts.

Before diving into the specific questions, let's recap the basics of normal distribution. The normal distribution is characterized by its bell shape, and it's fully defined by two parameters: the mean (μ) and the standard deviation (σ). The mean represents the average value of the dataset, while the standard deviation measures the spread or dispersion of the data around the mean. A smaller standard deviation indicates that the data points are clustered closely around the mean, while a larger standard deviation indicates a wider spread.

The Empirical Rule, also known as the 68-95-99.7 rule, provides a useful guideline for understanding the spread of data in a normal distribution. It states that:

  • Approximately 68% of the data falls within one standard deviation of the mean (μ ± 1σ).
  • Approximately 95% of the data falls within two standard deviations of the mean (μ ± 2σ).
  • Approximately 99.7% of the data falls within three standard deviations of the mean (μ ± 3σ).

This rule helps us quickly estimate the proportion of data within certain ranges. For instance, in our case, with a mean of 12 and a standard deviation of 2, we can expect that about 68% of the data will fall between 10 and 14, and about 95% will fall between 8 and 16.

In this section, we focus on determining the quartiles of a variable that is normally distributed. Quartiles divide a dataset into four equal parts. The first quartile (Q1) is the value below which 25% of the data falls, the second quartile (Q2) is the median (50%), and the third quartile (Q3) is the value below which 75% of the data falls. To find these quartiles, we need to use the standard normal distribution (Z-distribution), which has a mean of 0 and a standard deviation of 1. We will use the Z-table or statistical software to find the Z-scores corresponding to the 25th and 75th percentiles, and then convert these Z-scores back to the original scale of our variable.

To find the quartiles of a normally distributed variable, we first need to understand what quartiles represent. Quartiles divide a dataset into four equal parts. The first quartile (Q1) represents the 25th percentile, the second quartile (Q2) represents the 50th percentile (which is also the median), and the third quartile (Q3) represents the 75th percentile. For a normal distribution, we can determine these quartiles using the Z-score, which indicates how many standard deviations a value is from the mean. The Z-score formula is given by: Z = (X - μ) / σ, where X is the value, μ is the mean, and σ is the standard deviation. To find Q1 and Q3, we need to find the Z-scores corresponding to the 25th and 75th percentiles, respectively. We can use a Z-table or a calculator to find these values. The Z-score for the 25th percentile is approximately -0.674, and the Z-score for the 75th percentile is approximately 0.674. Now, we can use the Z-score formula to convert these Z-scores back to the original scale. For Q1, we have -0.674 = (Q1 - 12) / 2. Solving for Q1, we get Q1 = 12 + (-0.674) * 2 = 10.652. For Q3, we have 0.674 = (Q3 - 12) / 2. Solving for Q3, we get Q3 = 12 + 0.674 * 2 = 13.348. Therefore, the quartiles are approximately Q1 = 10.652, Q2 = 12 (the mean), and Q3 = 13.348. These values divide the dataset into four equal parts, with 25% of the data falling below 10.652, 50% falling below 12, and 75% falling below 13.348. Understanding the quartiles helps us to get a sense of the distribution and spread of the data.

Moving on to the 90th percentile, this is the value below which 90% of the data falls. This is another measure of position within the distribution. Again, we'll use the Z-table or statistical software to find the Z-score corresponding to the 90th percentile, and then convert it back to the original scale. Interpreting this percentile will give us a clear understanding of where a specific value stands in relation to the rest of the data.

The 90th percentile is a statistical measure that indicates the value below which 90% of the data falls. In the context of a normal distribution, this means that 90% of the values are less than or equal to this percentile. To find the 90th percentile, we once again use the Z-score. We look up the Z-score that corresponds to the 90th percentile in a Z-table or use a calculator. The Z-score for the 90th percentile is approximately 1.282. Using the Z-score formula, we can convert this Z-score back to the original scale: 1.282 = (X - 12) / 2. Solving for X, we get X = 12 + 1.282 * 2 = 14.564. Therefore, the 90th percentile is approximately 14.564. This means that 90% of the possible values of the variable are less than or equal to 14.564. Interpretation of the 90th percentile is crucial in many real-world applications. For example, if this variable represents test scores, a score at the 90th percentile would mean that the individual scored higher than 90% of the test-takers. Similarly, in business, the 90th percentile could represent the revenue generated by the top 10% of salespersons. Understanding and interpreting percentiles like the 90th percentile provides valuable insights into the distribution of data and the relative standing of individual values within that distribution. It helps in making informed decisions and comparisons across different datasets.

Here, we need to find the value that 65% of all possible values of the variable exceed. This is the same as finding the value below which 35% of the data falls (100% - 65% = 35%). So, we're essentially looking for the 35th percentile. We'll follow the same procedure as before, finding the Z-score for the 35th percentile and converting it back to the original scale. This type of problem is useful in scenarios where we want to know a cutoff point for a certain percentage of the population.

In this section, we address the question of finding the value that 65% of all possible values of the variable exceed. This is equivalent to finding the value below which 35% of the data falls, since 100% - 65% = 35%. Therefore, we are looking for the 35th percentile. To find this percentile, we again utilize the Z-score. We consult a Z-table or use a calculator to find the Z-score corresponding to the 35th percentile. The Z-score for the 35th percentile is approximately -0.385. This negative Z-score indicates that the value is below the mean. We then use the Z-score formula to convert this Z-score back to the original scale: -0.385 = (X - 12) / 2. Solving for X, we get X = 12 + (-0.385) * 2 = 11.23. Thus, the value that 65% of all possible values of the variable exceed is approximately 11.23. This means that 65% of the data points in this normal distribution are greater than 11.23. Understanding this concept is particularly useful in scenarios where we need to establish a cutoff point. For instance, if we are assessing student performance, we might want to identify the score that 65% of the students exceeded. This can help in setting benchmarks and identifying students who may need additional support. Similarly, in quality control, we might want to know the value that 65% of our products exceed to ensure a certain level of performance or durability. The ability to determine such values is a powerful tool in data analysis and decision-making.

Let's delve into a more comprehensive discussion about the practical applications and interpretations of the concepts we've explored. Normal distribution is a fundamental concept in statistics with widespread applications across various fields. Understanding quartiles, percentiles, and the ability to find values corresponding to specific proportions of data is essential for data analysis and decision-making.

In this final section, we delve into a more comprehensive discussion about the practical applications and interpretations of the concepts we've explored throughout this article. Normal distribution is a fundamental concept in statistics, with widespread applications across various fields, ranging from natural sciences to social sciences and even business and finance. The ability to understand and apply the principles of normal distribution, including the calculation and interpretation of quartiles, percentiles, and values corresponding to specific proportions of data, is essential for effective data analysis and informed decision-making. In practical terms, the concepts we've discussed allow us to make sense of data in a meaningful way. For instance, understanding quartiles helps us divide a dataset into four equal segments, providing insights into the spread and central tendency of the data. This is particularly useful in identifying outliers and understanding the distribution's shape. Percentiles, on the other hand, offer a more granular view, allowing us to determine the relative standing of a particular value within the dataset. This is invaluable in scenarios where we need to compare individual data points or assess performance against a benchmark. For example, in educational testing, percentiles are used to understand how a student's score compares to those of other students. Similarly, in healthcare, percentiles are used to track a child's growth relative to their peers. The ability to find values that exceed a certain percentage of the data is also crucial in various contexts. In manufacturing, this might involve determining the quality threshold that a certain percentage of products must meet. In finance, it could involve identifying investment opportunities that have a high probability of exceeding a certain return. Moreover, the normal distribution and its related concepts play a critical role in statistical inference, which involves making generalizations about a population based on a sample of data. By understanding the properties of the normal distribution, we can construct confidence intervals, conduct hypothesis tests, and make predictions about future observations. This is fundamental to scientific research, where researchers use statistical methods to draw conclusions from experimental data. In conclusion, a strong grasp of normal distribution and its applications is essential for anyone working with data. It empowers us to extract valuable insights, make informed decisions, and solve real-world problems across a wide range of disciplines. The exercises we've worked through in this article provide a solid foundation for further exploration and application of these concepts.

In conclusion, understanding and applying the concepts of normal distribution, quartiles, percentiles, and Z-scores are crucial for statistical analysis and interpretation. By working through these problems, we've gained a practical understanding of how to determine these values and what they mean in the context of a normally distributed variable. These skills are valuable in a variety of fields, allowing us to make informed decisions based on data. The normal distribution is a powerful tool, and mastering its properties opens doors to deeper insights and more effective problem-solving.