a. Find P(X > 6.5). b. Find P(5.5 <= X <= 7.5).
Answer :
Given :
The population mean $(μ) = 2.1$
The population standard deviation $(σ) = 3$
Solution :
a) The probability that x is more than 6.5 :
$$\text{P}(x > 6.5) = \text{P}\left(\frac{x – \mu}{\sigma} > \frac{6.5 – 2.1}{3}\right)$$ $$= \text{P}(z > 1.47)$$ $$= 1 – \text{P}(z < 1.47)$$ $$= 1 – 0.9292$$ $$= 0.0708$$
b) the probability that x is between 5.5 and 7.5 ( including ) :
$$\text{P}(5.5 \leq x \leq 7.5) = \text{P}\left(\frac{5.5 – 2.1}{3} \leq \frac{x – \mu}{\sigma} \leq \frac{7.5 – 2.1}{3}\right)$$ $$= \text{P}(1.13 \leq z \leq 1.8)$$ $$= \text{P}(z \leq 1.8) – \text{P}(z < 1.13)$$ $$= 0.9641 – 0.8708$$ $$= 0.0933$$
