Section 5.3: Applications of Normal Distributions

1. Sketch the density histogram for a normal random variable \(X\) with mean \(\mu\) and standard deviation \(\sigma\).
2. Convert a general (nonstandard) normal random variable \(X\) with mean \(\mu\) and standard deviation \(\sigma\) to a standard normal random variable \(Z\).
3. Explain how the conversion \(Z = \dfrac{X - \mu}{\sigma}\) standardizes a nonstandard normal random variable \(X\) with mean \(\mu\) and standard deviation \(\sigma\).
4. Sketch the area under the density histogram of a nonstandard normal random variable \(X\) corresponding to \(P(X \leq x)\), \(P(X \ \geq x)\), and \(P(a \leq X \leq b)\).
5. For a given value of \(x\), calculate \(P(X \leq x)\) or \(P(X \ \geq x)\) by standardizing \(X\) and using a table like Table A–2 from Triola and Triola.
6. For given values of \(a\) and \(b\), compute \(P(a \leq X \leq b)\) by standardizing \(X\) and using a table like Table A–2 from Triola and Triola.
7. Given a probability \(p\), find the value of \(x\) such that \(P(X \leq x) = p\).