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 \(5 \%\) for a \(N(10,2)\) distribution converted to a standard normal distribution

Short answer

The upper 5% point for a N(10,2) distribution corresponds to \( X = 13.29 \) in the original distribution and \( Z = 1.645 \) in the standard normal distribution.

Step-by-step solution

Understand the given distribution

The given distribution is a normal distribution \(N(10,2)\). Here, 10 is the mean (\( \mu \)) and 2 is the standard deviation (\( \sigma \)). We want to convert an area in this distribution to an equivalent area in a standard normal distribution.

Setting boundaries

The upper 5% i.e. percentile 95% is asked in the question. To get this percentile from a N(10,2) distribution, we will use the Z-table. The Z-score corresponding to the upper 5% area under the curve (or percentile 95%) from Z-table is approximately 1.645.

Convert the Z-score to an X-value in the given distribution

To convert the Z-score to an X-value in the original N(10,2) distribution, we use the formula \( X = Z* \sigma + \mu \). Here, \( Z = 1.645 \), \( \sigma = 2 \), and \( \mu = 10 \). Thus, substituting these values leads to \( X = 1.645*2 + 10 = 13.29 \)

Sketch the normal distributions

Finally, we can sketch the two normal distributions. The original N(10,2) will be a bell-shaped curve centered at 10, while the standard normal distribution will be a similar curve centered at 0. Mark the point \( X = 13.29 \) on the first graph and the point \( Z = 1.645 \) on the second. Shade the area to the right of these points, representing the upper 5% of the distribution.