Question

In: Statistics and Probability

A. A normal random variable has an unknown mean μ and a standard deviation σ =...

A. A normal random variable has an unknown mean μ and a standard deviation σ = 2. If the probability that x exceeds 6.5 is .9732; find μ.

B. A standard normal random variable has μ = 0 and a standard deviation σ = 1. Find the probability of less than -2.73.

C. A standard normal random variable has μ = 0 and a standard deviation σ = 1. Find the probability greater than 3.28.

D. A standard normal random variable has μ = 0 and a standard deviation σ = 1. Find the probability between -1.00 and 3.20.

Expert Solution

$\mathbf{Solution}$

$\mathbf{formula:}\;Z=\frac{X-\mu}{\sigma}$

${\color{Blue} \mathbf{A.}}\;\mathbf{given}:\sigma=2$

$\Rightarrow P(X>6.5)=0.9732$

$\Rightarrow P\left ( \frac{X-\mu}{\sigma}>\frac{6.5-\mu}{2} \right )=0.9732$

$\Rightarrow P\left ( Z>\frac{6.5-\mu}{2} \right )=0.9732$

From Z-table, Lookup for Z-value corresponding to area 0.9732 to the right of the normal curve.

$\Rightarrow P(Z>\mathbf{-1.93})=0.9732$

$-1.93=\frac{6.5-\mu}{2}$

$-3.86=6.5-\mu$

$\mu=6.5+3.86$

$\mu=\mathbf{{\color{Red} 10.36}}$

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${\color{Blue} \mathbf{B.}}\;$

$\Rightarrow P(Z<-2.73)=\mathbf{{\color{Red} 0.0032}}\;\;(\mathbf{from\;z-table})$

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${\color{Blue} \mathbf{C.}}\;$

$\Rightarrow P(Z>3.28)=\mathbf{{\color{Red} 0.0005}}\;\;(\mathbf{from\;z-table})$

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${\color{Blue} \mathbf{D.}}\;$

$\Rightarrow P(-1.00<Z<3.20)$

$\Rightarrow P(Z<3.20)-P(Z<-1.00)$

$\Rightarrow 0.9993-0.1587\;(\mathbf{from\;z-table})$

$\Rightarrow \mathbf{{\color{Red} 0.8406}}$

orchestra answered 2 years ago

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A. A normal random variable has an unknown mean μ and a standard deviation σ =...

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