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 upper \(30 \%\) for a \(N(48,5)\) distribution converted to a standard normal distribution

Short answer

The value corresponding to the upper 30% for a N(48,5) distribution on a standard normal distribution is approximately 51.6.

Step-by-step solution

Understand the Basics

First, it's important to understand that a normal distribution is symmetrical and bell-shaped, characterized by two parameters: the mean (\(μ\)) and the standard deviation (\(σ\)). The standard normal distribution has a mean of 0 and a standard deviation of 1.

Find Z-score for the given percentile

In this case, we need to find the z-score for the 70th percentile (because the upper 30% is equivalent to the 70th percentile from the bottom) of the standard normal distribution. Using a z-table or a statistics calculator, we find that the z-score is approximately 0.52.

Convert the Z-Score to the normal distribution score

Now, the z-score needs to be converted to the given normal distribution which is N(48,5). The formula to convert a z-score to a score in a given normal distribution is \(X = μ + z*σ\). In this case, \(μ = 48\) and \(σ = 5\), plugging our z-score into the equation gives \(X = 48 + 0.52*5 = 51.6\)

Sketch the Normal Distributions

Label the x-axis with the equivalents for the standard normal distribution and the given normal distribution. In this case, 0.52 on the standard normal is equivalent to 51.6 on N(48,5). Shade the area under both curves which stands for the upper 30% of the distributions.

Key concepts

Z-Score

The z-score, also known as the standard score, is a crucial concept in statistics that represents the number of standard deviations a data point is from the mean. Essentially, it's a measure of how far away a particular value is from the average, when compared across different normal distributions with varying means and standard deviations. To calculate the z-score of a value, use the formula:
\[ z = \frac{{X - \mu}}{{\sigma}} \]
Where \(X\) is the value in question, \(\mu\) is the mean of the distribution, and \(\sigma\) is the standard deviation. This transformation allows us to compare scores from different normal distributions on a standard scale, the standard normal distribution. Understanding the z-score is fundamental in converting percentages or areas from one normal distribution to an equivalent in another, as was required in the exercise provided.

Standard Normal Distribution

The standard normal distribution, which is a special case of the normal distribution, has a mean of 0 and a standard deviation of 1. All normal distributions can be converted to this standard scale using z-scores. The beauty of the standard normal distribution lies in its universality; no matter the original mean or deviation, after converting to z-scores, we use the same standard normal distribution for calculations. This is particularly useful for finding percentiles or probabilities corresponding to different values.
For the conversion in our exercise, after finding the z-score for the 70th percentile, we interpret it on the standard normal distribution. Tools like z-tables or statistical software often provide these values, allowing quick conversions and comparisons.

Percentile

A percentile is a measure used in statistics indicating the value below which a given percentage of observations in a group fall. For example, the 90th percentile is the value below which 90% of the data points can be found. In the context of normal distributions, percentiles represent areas under the curve to the left of a given z-score value. The exercise mentioned asked for the upper 30%, which corresponds to the 70th percentile from the bottom of the distribution.
Percentiles are often used to interpret test scores, measurements, and other data, providing a clear way to understand a particular score's position relative to the rest. Converting between percentiles and z-scores allows us to use the standard normal distribution to make this process standardized and efficient.

Mean and Standard Deviation

Mean and standard deviation are the two parameters that define a normal distribution. The mean (denoted as \(\mu\)) is the average value of all the data points in the distribution. In contrast, the standard deviation (denoted as \(\sigma\)) measures the dispersion or variability around the mean; it quantifies how spread out the values are.
To tailor scores from any normal distribution to a standard normal distribution for comparison, we adjust these values using the mean and standard deviation of the distribution in question. As done in the exercise, once we have a z-score, we can determine the corresponding value in any other normal distribution by rearranging the z-score formula as follows:
\[ X = \mu + z \times \sigma \]
This formula is vital for converting between distributions, which the exercise demonstrated by converting a z-score into a specific value for the given distribution with its particular mean and standard deviation.