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 endpoint for the standard normal distribution value 2.1 on a \(N(500,80)\) distribution is 668.

Step-by-step solution

Conversion of z-score

Firstly, you need to convert the given data from the standard normal distribution to a z-score, which is a measure of how many standard deviations an element is from the mean. For a standard normal distribution, a value of 2.1 would be a z-score of 2.1.

Application of z-score formula

Secondly, you need to apply the z-score formula to the \(N(500,80)\) distribution. The formula is \(Z = \frac{(X - \mu)}{ \sigma}\), where \(\mu\) is the mean and \(\sigma\) is the standard deviation. For the given distribution \(N(500,80)\), \(\mu = 500\) and \(\sigma = 80\). Solving for X, the formula becomes \(X = Z \cdot \sigma + \mu\). Substituting Z=2.1, \(\sigma=80\), and \(\mu=500\), we get \(X = 2.1 \cdot 80 + 500 = 668\).

Shading the regions

Lastly, you should sketch a graph of both distributions and shade in the matching regions. For the standard normal distribution, shade in the area to the right of z = 2.1. For the \(N(500,80)\) distribution, shade in the area to the right of X = 668.