Question

Find the specified areas for a normal density. (a) The area below 0.21 on a \(N(0.3,0.04)\) distribution (b) The area above 472 on a \(N(500,25)\) distribution (c) The area between 8 and 10 on a \(N(15,6)\) distribution

Short answer

The answers after calculations would respectively appear as percentages. Due to different normal distributions and specific areas asked for, there can be different results.

Step-by-step solution

Solving for part (a)

Initially calculate the Z-score using the standard normal variable formula Z = (X - μ)/σ. Here, X = 0.21, μ = 0.3 and σ = \sqrt{0.04}. After obtaining the Z score, consult a z-table or use a statistical calculator to find the area below the curve up to the calculated Z score value. Convert this to a percentage for the final answer.

Solving for part (b)

Next, calculate the Z-score for part (b) the same way as in step 1. However, with X = 472, μ = 500 and σ = \sqrt{25}. Due to the question asking for the area above 472, subtract the area up to the calculated Z-score (from z-table or statistical calculator) from 1. This provides the area to the right of Z or above 472.

Solving for part (c)

In order to find the area between two values, start by calculating the Z-scores for both values: For X1 = 8 and X2 = 10, μ = 15 and σ = \sqrt{6}. Afterwards, subtract the area up to Z1 (where Z1 is derived from X1=8) from the area up to Z2 (where Z2 is derived from X2=10). This yields the area between these Z-scores equivalent to the area between 8 and 10.