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

Answer:

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

Step-by-step 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

$z=\frac{x-\mu }{\sigma }=\frac{60-47}{6.3}=2.063492$

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

$z=\frac{x-\mu }{\sigma }=\frac{50-47}{6.3}=0.4761904$

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

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