Finding X For A Given Probability In Normal Distribution With Mean 250 And Standard Deviation 80
In the realm of statistics, the normal distribution, often called the Gaussian distribution, is a cornerstone concept. Its bell-shaped curve elegantly models numerous natural phenomena, from heights and weights to test scores and measurement errors. Understanding the properties of normal distributions is crucial for making informed decisions and drawing meaningful conclusions from data. In this article, we will explore how to find the value of X for a given probability in a normal distribution, using a specific example to illustrate the process.
Defining the Normal Distribution
The normal distribution is characterized by two key parameters: the mean (μ) and the standard deviation (σ). The mean represents the average value of the distribution, while the standard deviation measures the spread or dispersion of the data around the mean. A larger standard deviation indicates a wider spread, while a smaller standard deviation indicates a tighter clustering around the mean.
In our case, we are given that X is normally distributed with a mean (μ) of 250 and a standard deviation (σ) of 80. This means that the average value of X is 250, and the data points are spread out around this mean with a standard deviation of 80.
Understanding Probability and Cumulative Distribution Function (CDF)
The probability associated with a normal distribution represents the likelihood of observing a value within a specific range. The cumulative distribution function (CDF) provides the probability that a random variable X will take on a value less than or equal to a given value x. In mathematical notation, the CDF is denoted as P(X ≤ x).
In our problem, we are given that P(X ≤ x) = 0.9394. This means that there is a 93.94% probability that the random variable X will take on a value less than or equal to x. Our goal is to find the value of x that corresponds to this probability.
Using the Z-score to Find X
To find the value of x, we need to utilize the concept of the Z-score. The Z-score measures the number of standard deviations a particular value is away from the mean. It allows us to standardize a normal distribution, transforming it into a standard normal distribution with a mean of 0 and a standard deviation of 1.
The formula for calculating the Z-score is:
Z = (x - μ) / σ
where:
- x is the value we are interested in
- μ is the mean of the distribution
- σ is the standard deviation of the distribution
To find the value of x that corresponds to a given probability, we first need to find the Z-score associated with that probability. We can do this using a Z-table or a statistical calculator.
Delving Deeper into Z-scores and Their Significance
To truly grasp the concept of Z-scores, it's essential to understand their fundamental role in standardizing normal distributions. A Z-score acts as a bridge, connecting any normal distribution to the standard normal distribution, a distribution with a mean of 0 and a standard deviation of 1. This standardization is pivotal because it allows us to utilize readily available Z-tables or statistical software to determine probabilities associated with any normal distribution.
Imagine you have two datasets, one representing the heights of students in a school and the other representing the scores on a standardized test. These datasets might have different means and standard deviations, making direct comparisons challenging. This is where Z-scores come to the rescue. By converting each data point into a Z-score, we essentially express it in terms of how many standard deviations it lies away from the mean of its respective distribution.
A positive Z-score indicates that the data point is above the mean, while a negative Z-score signifies that it's below the mean. The magnitude of the Z-score tells us how far away the data point is from the mean in terms of standard deviations. For instance, a Z-score of 2 means the data point is 2 standard deviations above the mean, while a Z-score of -1.5 indicates it's 1.5 standard deviations below the mean.
The beauty of Z-scores lies in their ability to transform any normal distribution into a standard normal distribution. This standardization enables us to use Z-tables, which provide the cumulative probabilities for the standard normal distribution. These tables list the probability that a standard normal variable will be less than or equal to a given Z-score. By looking up the Z-score in the table, we can directly find the corresponding probability.
In our problem, we're given a probability of 0.9394 and our mission is to find the corresponding value of x in the normal distribution with a mean of 250 and a standard deviation of 80. The first step is to find the Z-score associated with this probability. We can achieve this by consulting a Z-table or employing a statistical calculator. These tools essentially reverse the lookup process, allowing us to find the Z-score that corresponds to a specific cumulative probability.
Once we have the Z-score, we can use the Z-score formula to solve for x. This formula, as we've seen, expresses the relationship between x, the mean (μ), the standard deviation (σ), and the Z-score. By plugging in the values for Z, μ, and σ, we can isolate x and determine its value.
Finding the Z-score for P(X ≤ x) = 0.9394
Looking up 0.9394 in a Z-table, we find that the corresponding Z-score is approximately 1.55. This means that the value x is 1.55 standard deviations above the mean.
Calculating X
Now that we have the Z-score, we can use the Z-score formula to solve for x:
- 55 = (x - 250) / 80
Multiplying both sides by 80, we get:
- 55 * 80 = x - 250
124 = x - 250
Adding 250 to both sides, we get:
x = 374
Therefore, the approximate value of x such that P(X ≤ x) = 0.9394 is 374.
The Importance of Precision in Statistical Calculations
In the realm of statistics, precision reigns supreme. The accuracy of our calculations directly impacts the reliability of our conclusions and decisions. Therefore, it's paramount to approach statistical problems with meticulous attention to detail and a commitment to minimizing errors.
In our quest to find the value of x for a given probability in a normal distribution, we've encountered the critical step of using a Z-table or statistical calculator to determine the Z-score. This step underscores the importance of precision because Z-tables provide probabilities for specific Z-scores, and even slight deviations in the Z-score can lead to noticeable differences in the corresponding probability.
Imagine we were to use an approximate Z-score of 1.5 instead of the more precise 1.55. This seemingly small difference could result in a significant error in our final calculation of x. The cumulative probability associated with a Z-score of 1.5 is different from that associated with 1.55, and this difference would propagate through the subsequent steps, leading to an inaccurate value for x.
To mitigate the risk of errors, it's essential to use Z-tables or statistical calculators with sufficient precision. Most Z-tables provide probabilities to two decimal places, and statistical calculators can often provide even greater precision. When looking up Z-scores, it's crucial to use as many decimal places as possible to minimize rounding errors.
In addition to using precise Z-scores, it's equally important to maintain precision throughout the rest of the calculation. When performing multiplications, divisions, additions, and subtractions, avoid rounding intermediate results until the final step. Rounding intermediate values can introduce errors that accumulate and affect the accuracy of the final answer.
Moreover, it's wise to double-check your calculations to ensure that no mistakes have been made. Even a simple arithmetic error can lead to a drastically different result. By carefully reviewing your work, you can catch and correct any errors before they impact your conclusions.
The significance of precision extends beyond academic exercises. In real-world applications of statistics, where decisions are made based on data analysis, the consequences of inaccurate calculations can be substantial. For example, in medical research, an incorrect calculation could lead to the misinterpretation of clinical trial results, potentially affecting patient care. In finance, errors in statistical analysis could lead to poor investment decisions, resulting in financial losses.
Therefore, embracing precision in statistical calculations is not merely an academic pursuit; it's a professional responsibility. By striving for accuracy in our calculations, we ensure the reliability of our findings and contribute to sound decision-making in various fields.
Conclusion
In this article, we have explored the process of finding the value of X for a given probability in a normal distribution. We have seen how the Z-score plays a crucial role in standardizing normal distributions and allowing us to use Z-tables or statistical calculators to find probabilities. By understanding these concepts, we can effectively analyze and interpret data that follows a normal distribution.
Therefore, the correct answer is C. 374
