Let X be normally distributed with mean $\mu =2.1$ and standard deviation $\sigma =3$.

a. Find P(X > 6.5). b. Find P(5.5 <= X <= 7.5).

Answer :

Given :

The population mean $(\mu )=2.1$

The population standard deviation $(\sigma )=3$

Solution :

a) The probability that x is more than 6.5 :

$$\text{P}(x>6.5)=\text{P}(\frac{x–\mu }{\sigma }>\frac{6.5–2.1}{3})$$ $$=\text{P}(z>1.47)$$ $$=1–\text{P}(z<1.47)$$ $$=1–0.9292$$ $$=0.0708$$

b) the probability that x is between 5.5 and 7.5 ( including ) :

$$\text{P}(5.5\le x\le 7.5)=\text{P}(\frac{5.5–2.1}{3}\le \frac{x–\mu }{\sigma }\le \frac{7.5–2.1}{3})$$ $$=\text{P}(1.13\le z\le 1.8)$$ $$=\text{P}(z\le 1.8)–\text{P}(z<1.13)$$ $$=0.9641–0.8708$$ $$=0.0933$$