Question

Compute the following probabilities:
a. If Y is distributed N (-8,9), Pr (Y≤-12) = —— . (Round your response to four decimal places.)
b. If Y is distributed N (-4,9), Pr (Y>-1) = ——. (Round your response to four decimal places.)
c. If Y is distributed N (40,25), Pr (34≤Y≤44) = ——. (Round your response to four decimal places.)

Answer 1

Answer: a. 0.3300   b. 0.3707 c. 0.1584

Explanation:

If X is distributed N($\mu,\sigma^2$)  [Normal distribution], then Z-score = $\dfrac{X-\mu}{\sigma}$ .

a. If Y is distributed N (-8,9),

Pr (Y≤-12) $=P(\dfrac{Y-\mu}{\sigma}\leq \dfrac{-12-(-8)}{\sqrt{9}})$

$=P(Z\leq -1.33)=1-P(Z<1.33)\\\\=1- 0.9082\ \ \ [\text{By z-table}]\\\\= 0.0917$

b.  If Y is distributed N (-4,9),

Pr (Y>-1) =$=P(\dfrac{Y-\mu}{\sigma}>\dfrac{-1-(-4)}{3})$

$=P(Z>1)=1-P(Z<1)\\\\=1-0.8413\ \ \ [\text{By z-table}]\\\\=0.1587$

c. If Y is distributed N (40,25),

Pr (34≤Y≤44) = $P(\dfrac{34-40}{5}<\dfrac{Y-\mu}{\sigma}<\dfrac{44-40}{5})$

$P(-1.2<Z<0.8)=P(Z<0.8)-P(Z<-1.2)\\\\=P(Z<0.8)-(1-P(Z<1.2))\\\\=0.7881-(1-0.8849)\ \ \ [\text{By z-table}]\\\\=0.6730$