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 2.1 for a standard normal distribution converted to a \(N(500,80)\) distribution

Short answer

The equivalent area in the \(N(500,80)\) distribution lies above the calculated x-value.

Step-by-step solution

Identify the given information

The exercise provides the information that the area lies above 2.1 in the standard normal distribution and conversions need to be made for a different normal distribution \(N(500,80)\).

Convert z-value to x-value

To convert from the standard normal distribution to a different normal distribution, use the formula: \(x = z*SD + mean\). In this, z is the value from the standard normal distribution, SD is the standard deviation, and mean is the mean of the new normal distribution. Substitute the values, \(x = 2.1*80 + 500\) to find x.

Label and Shade the Regions on the graph

Draw two number lines, one for the standard normal distribution and the other for the \(N(500,80)\) distribution. For the standard normal distribution, shade the area above 2.1. After finding the x-value in the previous step, for the \(N(500,80)\) distribution, shade the area above the calculated x-value. The shaded area under both the distributions represent the same probability areas.

Key concepts

Standard Normal Distribution

The standard normal distribution is a special case of the normal distribution with a mean of 0 and a standard deviation of 1. It's represented by the symbol Z and is sometimes called the Z-distribution. Because of its standardization, it allows for the comparison of scores from different normal distributions.

Essentially, this distribution serves as a reference point, and with the help of z-score calculations, we can translate scores from any normal distribution to the standard normal distribution. This is particularly useful because the standard normal distribution has a well-defined table, known as the Z-table, which lists the probability of a score being below a particular value of Z.

One of the most significant properties of the standard normal distribution is its symmetry about the mean. Half of the scores lie below the mean, and half lie above. This symmetry makes it easier to understand the probabilities associated with scores that fall within certain standard deviations from the mean.

Z-Score Calculation

A z-score represents the distance a data point is from the mean, measured in standard deviations. It's a way to standardize scores from any normal distribution so you can understand where they fall within the standard normal distribution. To calculate a z-score, use the formula: \(z = \frac{x - \text{mean}}{\text{SD}}\), where x is the data point, mean is the average of the distribution, and SD is the standard deviation.

By converting a raw score to a z-score, we effectively 'normalize' it, enabling us to make meaningful comparisons between data points from different normal distributions or to assess the relative standing of a data point within its distribution. This process is crucial when dealing with data that doesn't initially fit the standard normal distribution but needs to be assessed using its properties.

Probability Areas

Probability areas within a normal distribution signify the likelihood of a variable falling within a certain range. These areas under the curve of the distribution represent cumulative probabilities. For example, in a standard normal distribution, approximately 68% of scores fall within one standard deviation of the mean, and about 95% within two standard deviations.

The significance of identifying these areas comes into play when predicting events or assessing the chance of a score occurring. In practice, this allows us to understand the proportion of the population that lies above or below a certain value, or within an interval. For instance, in the context of our exercise, the area above a z-score of 2.1 indicates a certain probability, which we aim to translate meaningfully to another normal distribution.

Normal Distribution Properties

Normal distribution is characterized by a bell-shaped curve, ubiquitously found in the realm of statistics and probability. Some of its key properties include its symmetry about the mean, with the tails of the curve approaching but never touching the horizontal axis, reflecting that extreme values are possible but improbable.

The total area under the curve equals 1, as it represents the entire range of possible outcomes. Furthermore, it's defined entirely by its mean and standard deviation, which tells us the central tendency and the dispersion of the data, respectively. The mean, median, and mode of a perfectly normal distribution are all equal, which is not necessarily true for distributions that are merely 'normal-like' or skewed.

Understanding these properties is crucial when performing various statistical analyses, as many statistical tests assume or require normally distributed data. Thus, appreciating the standard features of the normal distribution allows us to properly apply and interpret statistical procedures.