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 below 40 for a \(N(48,5)\) distribution converted to a standard normal distribution

Short answer

The Z score corresponding to '40' in our original normal distribution \(N(48,5)\) is \((40-48)/5\). Therefore, the conversion from the area below 40 in the \(N(48,5)\) distribution to a standard normal distribution involves finding this Z score and shading the area below this value on a standard normal distribution graph.

Step-by-step solution

Identify Original Mean and Standard Deviation

The normal distribution \(N(48,5)\) provided in the problem has a mean (mu) of 48 and a standard deviation (sigma) of 5.

Standardize The Given Value

To convert a score of 40 from distribution \(N(48,5)\) to standard distribution \(N(0,1)\), use the standardization formula \(Z = (X - mu) / sigma\). Here, X is the original score, mu is the mean, and sigma is the standard deviation. In this case, \(Z = (40 - 48) / 5\). This will give the equivalent 'Z' score value for '40' in standard normal distribution.

Sketch The Distributions and label the endpoints

Now, draw the original normal distribution \(N(48,5)\) with the area below 40 shaded and labelled. Then, draw the standard normal distribution \(N(0,1)\), and match the equivalent Z value we calculated and shade and label the area below this point.

Confirm The Areas Match

Confirm that the areas shaded in both graphs look similar, noting that the area under a probability curve represents the probability of events occurring. In theory and practice, the areas under the two curves must be the same as the standardization process doesn't change the probability, only the scale of the distribution.

Key concepts

Standard Normal Distribution

The standard normal distribution is a specific type of normal distribution that is particularly important in statistics. It's a bell-shaped curve that is symmetric about the mean, where the mean (also known as the expected value) is zero, and the standard deviation (a measure of spread) is one. This is denoted as \( N(0,1) \).

Due to its standardized nature, the standard normal distribution serves as a reference point and allows us to compare scores from different normal distributions. It is the basis for z-scores, which tell us how many standard deviations a data point is from the mean. Every normal distribution can be converted to the standard normal distribution using a process called standardization, which we'll explore next.

Standardization of Distribution

Standardization is the key process by which we can compare scores from different normal distributions. This involves converting scores from a normal distribution with any mean \( (\mu) \) and standard deviation \( (\sigma) \) to the standard normal distribution. The formula for standardization is given by:
\[ Z = \frac{{X - \mu}}{{\sigma}} \]
where \( X \) is the original score, \( \mu \) is the mean of the original normal distribution, and \( \sigma \) is its standard deviation.

By substituting the original values into this formula, we re-scale the distribution so that it has a mean of 0 and standard deviation of 1. This process does not alter the shape of the distribution or the relative standing of scores; it simply re-scales the axis to match that of the standard normal distribution.

Z-score Calculation

A Z-score measures the number of standard deviations a data point is from the mean of the standard normal distribution. To calculate the Z-score you would use the standardization formula mentioned previously. Let’s illustrate this with the example given in the original exercise.

First, calculate the Z-score for the value 40 from the normal distribution with mean 48 and standard deviation 5:
\[ Z = \frac{{40 - 48}}{{5}} = -1.6 \]
This Z-score tells us that 40 is 1.6 standard deviations below the mean of the original normal distribution. The negative sign indicates that 40 is less than the mean.

Remember, it’s essential to sketch the distributions before and after standardization. For the original distribution \( N(48,5) \), the point at 40 would be shaded to the left of the mean. On the standard normal distribution graph, the point corresponding to the Z-score of -1.6 would be shaded. These diagrams help ensure clarity by visualizing the comparability of areas under the curves, reaffirming that standardization preserves the probabilities.