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 middle \(80 \%\) for a standard normal distribution converted to a \(N(100,15)\) distribution

Short answer

The middle 80% of the data for a standard normal distribution that's converted into a N(100,15) distribution falls between 80.8 and 119.2.

Step-by-step solution

Identify the Area

The problem states that we are looking for the middle 80% of the distribution. This means we have 10% of the data on either side of the middle 80%. In a standard normal distribution, this corresponds to a z-value. We can find this z-value from a standard normal distribution table or from the Z-SCORE formula. We get a score of ±1.28

Convert to N(100,15)

Now, we convert these z-scores back into scores in the original distribution. The formula for conversion is \(X = \mu + Z\sigma\), where \(\mu\) represents the mean, \(Z\) the z-score, and \(\sigma\) the standard deviation. So for \(Z=1.28\), \(X = 100 + 1.28(15) = 119.2 ; Z=-1.28, X = 100 - 1.28(15) = 80.8 \).

Sketch the Normal Distributions

We sketch two normal distribution curves: one for the standard normal distribution and one for the N(100, 15) distribution. On both curves, we label the endpoints (±1.28 for the standard normal distribution and 80.8 and 119.2 for the N(100,15) distribution) and shade the region between them. This represents 80% of the data.