Answer:
(a) P(0 ≤ Z ≤ 2.87)=0.498
(b) P(0 ≤ Z ≤ 2)=0.477
(c) P(−2.20 ≤ Z ≤ 0)=0.486
(d) P(−2.20 ≤ Z ≤ 2.20)=0.972
(e) P(Z ≤ 1.01)=0.844
(f) P(−1.95 ≤ Z)=0.974
(g) P(−1.20 ≤ Z ≤ 2.00)=0.862
(h) P(1.01 ≤ Z ≤ 2.50)=0.150
(i) P(1.20 ≤ Z)=0.115
(j) P(|Z| ≤ 2.50)=0.988
Step-by-step explanation:
(a) P(0 ≤ Z ≤ 2.87)
In this case, this is equal to the difference between P(z<2.87) and P(z<0). The last term is substracting because is the area under the curve that is included in P(z<2.87) but does not correspond because the other condition is that z>0.
[tex]P(0 \leq z \leq 2.87)= P(z<2.87)-P(z<0)=0.998-0.5=0.498[/tex]
(b) P(0 ≤ Z ≤ 2)
This is the same case as point a.
[tex]P(0 \leq z \leq 2)= P(z<2)-P(z<0)=0.977-0.5=0.477[/tex]
(c) P(−2.20 ≤ Z ≤ 0)
This is the same case as point a.
[tex]P(-2.2 \leq z \leq 0)= P(z<0)-P(z<-2.2)=0.5-0.014=0.486[/tex]
(d) P(−2.20 ≤ Z ≤ 2.20)
This is the same case as point a.
[tex]P(-2.2 \leq z \leq 2.2)= P(z<2.2)-P(z<-2.2)=0.986-0.014=0.972[/tex]
(e) P(Z ≤ 1.01)
This can be calculated simply as the area under the curve for z from -infinity to z=1.01.
[tex]P(z\leq1.01)=0.844[/tex]
(f) P(−1.95 ≤ Z)
This is best expressed as P(z≥-1.95), and is calculated as the area under the curve that goes from z=-1.95 to infininity.
It also can be calculated, thanks to the symmetry in z=0 of the standard normal distribution, as P(z≥-1.95)=P(z≤1.95).
[tex]P(z\geq -1.95)=0.974[/tex]
(g) P(−1.20 ≤ Z ≤ 2.00)
This is the same case as point a.
[tex]P(-1.20 \leq z \leq 2.00)= P(z<2)-P(z<-1.2)=0.977-0.115=0.862[/tex]
(h) P(1.01 ≤ Z ≤ 2.50)
This is the same case as point a.
[tex]P(1.01 \leq z \leq 2.50)= P(z<2.5)-P(z<1.01)=0.994-0.844=0.150[/tex]
(i) P(1.20 ≤ Z)
This is the same case as point f.
[tex]P(z\geq 1.20)=0.115[/tex]
(j) P(|Z| ≤ 2.50)
In this case, the z is expressed in absolute value. If z is positive, it has to be under 2.5. If z is negative, it means it has to be over -2.5. So this probability is translated to P|Z| < 2.50)=P(-2.5<z<2.5) and then solved from there like in point a.
[tex]P(|z|<2.5)=P(-2.5<z<2.5)=P(z<2.5)-P(z<-2.5)\\\\P(|z|<2.5)=0.994-0.006=0.988[/tex]
Answer:
a) 0.49795
b) 0.47725
c) 0.4861
d) 0.9722
e) 0.84375
f) 0.02559
g) 0.86218
h) 0.15004
i) 0.11507
j) 0.9876
Step-by-step explanation:
attached is the statistical table for your reference.
(a) P(0 ≤ Z ≤ 2.87)
P( Z ≤ 2.87) - P(Z ≤ 0) ⇒From statistical table of positive z values
= 0.99795 - 0.5 = 0.49795
(b) P(0 ≤ Z ≤ 2)
P( Z ≤ 2) - P(Z ≤ 0) ⇒From statistical table of positive z values
= 0.97725 - 0.5 = 0.47725
(c) P(−2.20 ≤ Z ≤ 0)
P( Z ≤ 0) - P(Z ≤ -2.20) ⇒From statistical table of positive&negative z values
= 0.5 - 0.01390 = 0.4861
(d) P(−2.20 ≤ Z ≤ 2.20)
P( Z ≤ 2.20) - P(Z ≤ -2.20) ⇒From statistical table of positive&negative z values
= 0.98610 - 0.01390 = 0.9722
(e) P(Z ≤ 1.01) ⇒From statistical table of positive z values
0.84375
(f) P(−1.95 ≤ Z)
P(Z ≥ −1.95)
= 1 - P(Z < 1.95) ⇒From statistical table of positive z values
= 1 - 0.97441 = 0.02559
(g) P(−1.20 ≤ Z ≤ 2.00)
P( Z ≤ 2.00) - P(Z ≤ -1.20) ⇒From statistical table of positive&negative z values
= 0.97725 - 0.11507 = 0.86218
(h) P(1.01 ≤ Z ≤ 2.50)
P( Z ≤ 2.50) - P(Z ≤ 1.01) ⇒From statistical table of positive z values
= 0.99379 - 0.84375 = 0.15004
(i) P(1.20 ≤ Z)
P(Z ≥ 1.20)
= 1 - P(Z < 1.20) ⇒From statistical table of positive z values
= 1 - 0.88493 = 0.11507
(j) P(|Z| ≤ 2.50)
P(-2.50 ≤ Z ≤ 2.50)
P(Z ≤ 2.50) - P(Z ≤ -2.50) ⇒From statistical table of positive&negative z values
= 0.9938 - 0.0062
= 0.9876
