Question

Let Z be a standard normal random variable and calculate the following probabilities, drawing pictures wherever appropriate. (Round your answers to four decimal places.) (a) P(0 ≤ Z ≤ 2.87) .9979 Incorrect: Your answer is incorrect. (b) P(0 ≤ Z ≤ 2) .9772 Incorrect: Your answer is incorrect. (c) P(−2.20 ≤ Z ≤ 0) .0139 Incorrect: Your answer is incorrect. (d) P(−2.20 ≤ Z ≤ 2.20) .9633 Incorrect: Your answer is incorrect. (e) P(Z ≤ 1.01) .8438 Correct: Your answer is correct. (f) P(−1.95 ≤ Z) .0256 Incorrect: Your answer is incorrect. (g) P(−1.20 ≤ Z ≤ 2.00) .8621 Correct: Your answer is correct. (h) P(1.01 ≤ Z ≤ 2.50) 0.15 Correct: Your answer is correct. (i) P(1.20 ≤ Z) .8849 Incorrect: Your answer is incorrect. (j) P(|Z| ≤ 2.50) .9938 Incorrect: Your answer is incorrect.

Answer 1

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

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Answer 2

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.

$P(0 \leq z \leq 2.87)= P(z$

(b) P(0 ≤ Z ≤ 2)

This is the same case as point a.

$P(0 \leq z \leq 2)= P(z$

(c) P(−2.20 ≤ Z ≤ 0)

This is the same case as point a.

$P(-2.2 \leq z \leq 0)= P(z$

(d) P(−2.20 ≤ Z ≤ 2.20)

This is the same case as point a.

$P(-2.2 \leq z \leq 2.2)= P(z$

(e) P(Z ≤ 1.01)

This can be calculated simply as the area under the curve for z from -infinity to z=1.01.

$P(z\leq1.01)=0.844$

(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).

$P(z\geq -1.95)=0.974$

(g) P(−1.20 ≤ Z ≤ 2.00)

This is the same case as point a.

$P(-1.20 \leq z \leq 2.00)= P(z$

(h) P(1.01 ≤ Z ≤ 2.50)

This is the same case as point a.

$P(1.01 \leq z \leq 2.50)= P(z$

(i) P(1.20 ≤ Z)

This is the same case as point f.

$P(z\geq 1.20)=0.115$

(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.

$P(|z|$