Question

Given that x is a normal variable with mean μ = 47 and standard deviation σ = 6.3, find the following probabilities. (Round your answers to four decimal places.) (a) P(x ≤ 60) (b) P(x ≥ 50) (c) P(50 ≤ x ≤ 60)

Answer 1

(a)
`P\left ( x\leq 60 \right )=0.9803`, (b)
`P\left ( x\geq 50 \right )=0.3156`, (c)
`P(50 \leq x \leq 60)=0.2959`

Explanation:

Given that mean
`\mu =47` and standard deviation
`\sigma =6.3`

(a) We need to find
`P\left ( x\leq 60 \right )`. Using z score

`z=(x-\mu )/(\sigma )=(60-47)/(6.3)=2.063492`

So,

`P\left ( x\leq 60 \right )=P\left ( z\leq 2.063492 \right )=0.9803`

(b) We need to find
`P\left ( x\geq 50 \right )`. Using z score

`z=(x-\mu )/(\sigma )=(50-47)/(6.3)=0.4761904`

So,

`P\left ( x\geq 50 \right )=P\left ( z\geq 0.4761904 ) \right )=1-P\left ( z\leq 0.4761904 ) \right )=1-0.6844=0.3156`

(c) We need to find
`P(50 \leq x \leq 60)`. From above z scores

`P(50 \leq x \leq 60)=P\left ( 0.4761904\leq z\leq2.063492 \right )=0.2959`