Question

Ask you to convert an area from one normal distribution to an equivalent area for a different normal distribution. Draw sketches of both normal distributions, find and label the endpoints, and shade the regions on both curves. The area above 13.4 for a \(N(10,2)\) distribution converted to a standard normal distribution

Short answer

The equivalent area for the standard normal distribution \(N(0,1)\) corresponding to the area above 13.4 in the \(N(10,2)\) distribution is above the Z-score of 1.7. It is shown as the shaded area in the sketch of the \(N(0,1)\) distribution.

Step-by-step solution

Identify the Given Values

For the given normal distribution \(N(10,2)\), the mean \(\mu = 10\) and standard deviation \(\sigma = 2\). The value \(X = 13.4\) is the score we are interested in.

Insert the Values into the Conversion Formula

Apply a transformation to the score using the formula \(Z = (X - \mu)/\sigma\). Inserting the given values gives us \(Z = (13.4 - 10) / 2\).

Calculate the Z-Score

Evaluate the expression to find \(Z = 1.7\). This is the standard normal score (or Z-score) that corresponds to \(X = 13.4\) in the \(N(10,2)\) distribution.

Draw the Normal Distributions and Areas

Sketch the given \(N(10,2)\) distribution as well as the standard normal distribution \(N(0,1)\). In the \(N(10,2)\) distribution, mark the point \(X = 13.4\) and shade the area to the right of it. In the \(N(0,1)\) distribution, mark the point \(Z = 1.7\) and shade the area to the right of it, which represents the same proportion of the area as in the \(N(10,2)\) distribution. The shaded areas represent the same proportion of the population in both distributions.

Key concepts

Z-score Calculation

Understanding the concept of Z-score is pivotal when dealing with normal distributions. A Z-score, also referred to as a standard score, quantifies the number of standard deviations a data point is from the mean of the distribution. It's a key tool for comparing different data points within the same distribution or between different normal distributions. Here's how the Z-score is calculated in a real-world scenario.

To determine the Z-score, you use the formula \[\begin{equation} Z = \frac{(X - \mu)}{\sigma} \end{equation}\] where \begin{itemize}\item \( X \) is the value from the original distribution for which you're calculating the Z-score,\item \( \mu \) is the mean of the original distribution, and\item \( \sigma \) is the standard deviation of the original distribution. \end{itemize}In the context of our exercise, the value 13.4 from the distribution \( N(10,2) \) can be converted to a Z-score by subtracting the mean (10) and then dividing by the standard deviation (2). This results in a Z-score of 1.7, meaning that 13.4 is 1.7 standard deviations above the mean in this particular distribution.

Standard Normal Distribution

The standard normal distribution is a special case of the normal distribution and serves as a fundamental concept in statistics. It is a normal distribution with a mean (\( \mu \)) of 0 and a standard deviation (\( \sigma \)) of 1. The beauty of the standard normal distribution, often denoted as \( N(0,1) \), is that it provides a reference framework to which any normal distribution can be compared through the use of Z-scores.

Every Z-score you calculate corresponds to a specific point on the standard normal distribution. If, for example, you have a Z-score of 1.7, this would pinpoint a location on the standard normal curve that is 1.7 standard deviations above the mean. This allows for direct comparisons between different data and even different distributions, as we align them to a common scale—the standard normal distribution. Translating scores to this scale is akin to translating different languages into a single universal language, simplifying communication and comprehension across diverse datasets.

Normal Distribution Properties

Normal distributions are continuous probability distributions that are symmetric around their mean, characterized by their bell-shaped curves. They have some distinct properties which make them extremely useful across various fields, from psychology to finance. Here are a few key properties of normal distributions:

• The distribution is symmetric about the mean, meaning the left and right halves of the graph are mirror images.
• The mean, median, and mode of a normal distribution are all equal and located at the center of the distribution.
• Approximately 68% of the data within a normal distribution lies within one standard deviation of the mean, about 95% lies within two standard deviations, and nearly all (99.7%) lies within three standard deviations (this is known as the empirical rule).
• The tails of the normal distribution curve extend infinitely in both directions and never touch the x-axis, theoretically encompassing all possible outcomes.

These properties not only help in analytical computations but also assist in making sense of data by providing a structured approach to interpreting variability and probability within datasets.