Observations Between Z=-1.08 And Z=0.92 In A Standard Normal Distribution

Understanding the Standard Normal Distribution

In the realm of statistics, the standard normal distribution reigns supreme as a cornerstone concept. It's a probability distribution characterized by its symmetrical bell-shaped curve, with a mean of 0 and a standard deviation of 1. This distribution serves as a powerful tool for understanding and analyzing data, allowing us to determine the probability of observing values within a specific range. The standard normal distribution simplifies complex statistical analyses by providing a standardized framework. Its properties allow statisticians and researchers to easily calculate probabilities and make inferences about populations based on sample data. The applications of the standard normal distribution span across various fields, including finance, engineering, and social sciences, highlighting its versatility and importance. The beauty of the standard normal distribution lies in its ability to transform any normal distribution into a standard form, making comparisons and calculations straightforward. This standardization process involves converting raw data points into z-scores, which represent the number of standard deviations a particular data point is away from the mean. By utilizing z-scores, we can readily determine the probability associated with any value within a normal distribution using the z-table, a crucial resource for statistical analysis. The standard normal distribution not only provides a foundation for hypothesis testing and confidence interval estimation but also enhances our ability to interpret and communicate statistical findings effectively. Its role in statistical modeling and data analysis cannot be overstated, making it an essential concept for anyone working with quantitative data.

Decoding Z-Scores and the Z-Table

Z-scores act as the language of the standard normal distribution, quantifying how far a data point deviates from the mean. They are expressed in terms of standard deviations, providing a standardized measure of relative position within the distribution. A z-score of 0 indicates that the data point coincides with the mean, while positive and negative z-scores signify values above and below the mean, respectively. The magnitude of the z-score reflects the distance from the mean; for instance, a z-score of 2 implies that the data point is two standard deviations above the mean. To effectively utilize z-scores, we turn to the z-table, also known as the standard normal table. This table serves as a lookup directory, providing the cumulative probability associated with each z-score. The cumulative probability represents the proportion of observations falling below a given z-score in the standard normal distribution. The z-table is meticulously organized, typically displaying z-scores in increments of 0.01 along the rows and columns, allowing for precise probability determination. Navigating the z-table involves locating the desired z-score by identifying the row corresponding to the integer and first decimal place and the column corresponding to the second decimal place. The intersection of the row and column yields the cumulative probability. For instance, to find the probability associated with a z-score of 1.50, we would locate the row labeled 1.5 and the column labeled 0.00, revealing a cumulative probability of 0.9332. This implies that 93.32% of the observations in the standard normal distribution fall below a z-score of 1.50. The z-table empowers us to translate z-scores into probabilities and vice versa, enabling a deeper understanding of data distribution and facilitating statistical inference.

Solving the Problem: Observations Between z = -1.08 and z = 0.92

The problem at hand asks us to determine the percentage of observations that fall between z = -1.08 and z = 0.92 in a standard normal distribution. To solve this, we'll employ the z-table and the fundamental properties of the distribution. Our strategy involves finding the cumulative probabilities associated with each z-score and then calculating the difference to determine the probability within the specified range. First, we'll look up the cumulative probability for z = 0.92 in the z-table. Locating the row labeled 0.9 and the column labeled 0.02, we find a corresponding probability of approximately 0.8212. This means that 82.12% of observations lie below z = 0.92. Next, we'll find the cumulative probability for z = -1.08. The z-table typically displays probabilities for positive z-scores, but we can leverage the symmetry of the standard normal distribution. The probability of an observation falling below z = -1.08 is equal to 1 minus the probability of falling below z = 1.08. Looking up z = 1.08 in the z-table, we find a probability of approximately 0.8599. Therefore, the probability of falling below z = -1.08 is 1 - 0.8599 = 0.1401, or 14.01%. Now, to find the percentage of observations between z = -1.08 and z = 0.92, we subtract the cumulative probability of z = -1.08 from the cumulative probability of z = 0.92: 0.8212 - 0.1401 = 0.6811. This result indicates that approximately 68.11% of observations fall within the specified range. Therefore, the correct answer is C. 68.11%.

Choosing the Correct Answer

Based on our calculations, the percentage of observations falling between z = -1.08 and z = 0.92 in a standard normal distribution is approximately 68.11%. Now, let's revisit the provided options:

• A. 31.89%
• B. 46.13%
• C. 68.11%
• D. 53.87%

Comparing our calculated result with the options, we can clearly see that option C, 68.11%, matches our findings. The other options deviate significantly from our calculated percentage, making them incorrect. Option A, 31.89%, represents a much smaller percentage than what we calculated, suggesting a misunderstanding of the range or the use of the z-table. Option B, 46.13%, also falls short of our result, indicating a possible error in either the z-score lookup or the probability calculation. Option D, 53.87%, while closer to our answer than options A and B, still does not align with the accurate percentage we determined. Therefore, we can confidently affirm that the correct answer is C. 68.11%, as it accurately reflects the percentage of observations between z = -1.08 and z = 0.92 in a standard normal distribution.

Practical Applications of the Standard Normal Distribution

The standard normal distribution is not merely a theoretical construct; it's a versatile tool with widespread practical applications across diverse fields. Its ability to model various phenomena makes it indispensable in statistics, finance, engineering, and beyond. In finance, the standard normal distribution is a cornerstone of portfolio management and risk assessment. The distribution helps in modeling stock returns, option pricing, and other financial metrics, allowing investors to estimate probabilities of potential gains and losses. By understanding the distribution of financial assets, analysts can make informed decisions about investment strategies and risk mitigation. In engineering, the standard normal distribution plays a crucial role in quality control and reliability analysis. Engineers use the distribution to model variations in manufacturing processes, assess the reliability of products, and ensure that systems meet specified performance standards. Statistical process control, a key engineering discipline, heavily relies on the standard normal distribution to monitor production processes and identify potential deviations from desired specifications. The standard normal distribution also finds applications in medical research and public health. It is used to model the distribution of biological variables, such as blood pressure and cholesterol levels, in populations. This enables researchers to identify individuals at risk of certain diseases and evaluate the effectiveness of interventions. Epidemiological studies often employ the standard normal distribution to analyze disease patterns and trends, contributing to public health policy and interventions. Furthermore, the standard normal distribution is essential in hypothesis testing and statistical inference. Researchers use it to determine the significance of experimental results, draw conclusions about populations based on sample data, and make predictions about future events. The distribution's ability to standardize data and provide a framework for probability calculations makes it a fundamental tool for scientific research across various disciplines.