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.

Solutions

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.

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

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

$-3.86=6.5-\mu$

$\mu=6.5+3.86$

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

-----------------------------------------------------------------------------------------------------------

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

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

-----------------------------------------------------------------------------------------------------------

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

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

-----------------------------------------------------------------------------------------------------------

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

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

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

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

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

orchestra answered 1 year ago

Next > < Previous

Related Solutions

1) A normal random variable x has an unknown mean μ and standard deviation σ =...

1) A normal random variable x has an unknown mean μ and standard deviation σ = 2. If the probability that x exceeds 4.6 is 0.8023, find μ. (Round your answer to one decimal place.) μ = 2) Answer the question for a normal random variable x with mean μ and standard deviation σ specified below. (Round your answer to four decimal places.) μ = 1.3 and σ = 0.19. Find P(1.50 < x < 1.71). P(1.50 < x <...

A normal random variable x has mean μ = 1.8 and standard deviation σ = 0.18....

A normal random variable x has mean μ = 1.8 and standard deviation σ = 0.18. Find the probabilities of these X-values. (Round your answers to four decimal places.) (a) 1.00 < X < 1.50 (b) X > 1.39 (c) 1.45 < X < 1.60

The random variable X has mean μ and standard deviation σ. A simple random sample of...

The random variable X has mean μ and standard deviation σ. A simple random sample of 50 values drawn from X has sample mean 14.7. Assuming the standard deviation is known to be 3.6, construct an 98% confidence interval for the value of μ. Put your answer in correct interval notation, (lower bound, upper bound), and round to three decimal places.

1. Consider a standard normal random variable with μ = 0 and standard deviation σ =...

1. Consider a standard normal random variable with μ = 0 and standard deviation σ = 1. (Round your answers to four decimal places.) (a) P(z < 2) = (b) P(z > 1.16) = (c) P(−2.31 < z < 2.31) = (d) P(z < 1.82) = 2. Find the following probabilities for the standard normal random variable z. (Round your answers to four decimal places.) (a) P(−1.49 < z < 0.65) = (b) P(0.56 < z < 1.74) = (c) P(−1.54 < z < −0.46) = (d) P(z...

Question

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

Solutions

Expert Solution

Related Solutions

1) A normal random variable x has an unknown mean μ and standard deviation σ =...

A normal random variable x has mean μ = 1.8 and standard deviation σ = 0.18....

The random variable X has mean μ and standard deviation σ. A simple random sample of...

1. Consider a standard normal random variable with μ = 0 and standard deviation σ =...

1. Consider a standard normal random variable with μ = 0 and standard deviation σ =...

Consider a standard normal random variable with μ = 0 and standard deviation σ = 1....

Given that x is a normal variable with mean μ = 49 and standard deviation σ...

Given that x is a normal variable with mean μ = 43 and standard deviation σ...

Given that x is a normal variable with mean μ = 107 and standard deviation σ...

Given that x is a normal variable with mean μ = 49 and standard deviation σ...