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 ≤ 2.64) (b) P(0 ≤ Z ≤ 2) .4772 Correct: Your answer is correct. (c) P(−2.10 ≤ Z ≤ 0) (d) P(−2.10 ≤ Z ≤ 2.10) (e) P(Z ≤ 1.94) (f) P(−1.45 ≤ Z) (g) P(−1.10 ≤ Z ≤ 2.00) (h) P(1.94 ≤ Z ≤ 2.50) (i) P(1.10 ≤ Z) (j) P(|Z| ≤ 2.50) .9876 Correct: Your answer is correct.

Answer 1

(a) P(0 ≤ Z ≤ 2.64)

Step 1: Sketch the curve The probability that 0<Z< 2.64 is equal to the blue area under the curve. 0.5 0.4 0.3 y 0.2 0.1 по 2.64 -л. -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 0.5 1.0 15 2.0 2.5 3.0 3.5 Z<2.64 Step 2: To find the probability of P (0 we use the following formula: P (0Z2.64) P ( Z < 2.64) P (Z <0) Step 3: P (Z 2.64) can be found by using the following standard normal table. Show Standard Normal Table We see that P (Z<2.64) = 0,9959 Step 4: P (Z<0) can be found by using the following standard normal table. Show Standard Normal Table We see that P (Z<0)-0.5 At the end we have: P (0< Z<2.64)-0.4959

(b) P(0 ≤ Z ≤ 2) .

Step 1: Sketch the curve. The probability that 0<Z<2 is equal to the blue area under the curve. 0.5 0.4 y 0.2 0.1 0.0 2 -0.1 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Step 2: To find the probability of P (0< Z<2), we use the following formula: P (0 P(Z<2)-P (Z <0) Z2) Step 3: P (Z<2) can be found by using the following standard normal table. Show Standard Normal Table We see that P (Z<2)-0.9772 Step 4: P(Z0) can be found by using the following standard normal table. Show Standard Normal Table We see that P (Z0) 0.5. At the end we have: P (0 Z<2)- 0.4772

(c) P(−2.10 ≤ Z ≤ 0)

Step 1: Sketch the curve. The probability that 2.10 Z<0 is equal to the blue area under the curve. 0.5 0.4 0.3 0.2 0.1 0.0- -2.1 -0.1 -3.5 30 -7 5 -2.0 -1.5 .0 -05 0.0 0.5 1.0 1-5 2.0 2,5 3 0 3.5 Step 2: To find the probability of P(-2.10 Z0). we use the following formula: P-2.10 Z0) P(Z0)-P (Z < -2.10) Step 3: P(Z<0) can be found by using the following standard normal table. Show Standard Normal Table We see that P (Z<0)-0.5. = Step 4: P(Z<-2.10) can be found by using the following fomula. 1 - P(Z<a) P(Z-a) After substituting a 2.1 we have P(Z-2.10) =1 - P (Z< 2.1 P (Z< 2.1) can be found by using the following standard normal table. Show Standard Normal Table We see that P (Z<2.1)= 0.9821 so, P(Z-2.10) =1- 1-P(Z<2.1)-1-0.9821 0.0179 At the end we have P(-2.10Z 0) 0.4821

(d) P(−2.10 ≤ Z ≤ 2.10)

Step 1: Sketch the curve The probability that -2.10 < Z<2.10 is equal to the blue area under the curve 0.5 n 4 0.3 y 0.2 0.1 0.0 -2.1 2.1 -0.1 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 X Step 2: To find the probability of P(-2.10<Z< 2.10). we use the following formula: P (-2.10 Z< 2.10) P(Z 2.10) P (Z <-2.10) Step 3: P (Z<2.10) can be found by using the following standard normal table. Show Standard Normal Table We see that P (Z<2.10)0.9821. Step 4: P (Z< -2.10) can be found by using the following fomula. P(Z -a) 1- P(Z <a) After substituting a 2.1 we have: 1- P (Z<2.1) P(Z-2.10) can be found by using the following standard normal table. P(Z< 2.1) Show Standard Normal Table We see that P 0.9821 so, (Z<2.1) P(Z2.10)1 - 1- P(Z<2.1)= 1 0.9821 0.0179 At the end we have: P (-2.10Z<2.10) 0.9642

(e) P(Z ≤ 1.94)

Step 1: Sketch the curve. The probability that Z<1.94 is equal to the blue area under the curve. 0.5 0.4 0.3 y 0.2 0.0 1.94 -0.1 -3.5 -3.0 -2.5 -7.0 1.5 -1.0 -05 0.0 0.5 1.0 15 2.0 2.5 3.0 3.5 X Step 2: Use the standard normal table to find this area. Show Standard Normal Table Step 3: We conclude that P(Z< 1.94) 0.9738 un

P(−1.45 ≤ Z)

Step 1: Sketch the curve. The probability that Z>-1.45 is equal to the blue area under the curve 0.5 0.4- 0.3 0.2- y 0.1 0.0- -1.45 -п+ -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 X Step 2: Note that the area above -1.45 is same as area below 1.45. 0.5 0.4- 0.3 Y 0.2 0.1 0.0 1.45 -0.1 -3.5 -30 -2.5 2.0 -1.5 -1.0 -0.5 пл 0.5 1.0 15 2.0 2.5 3.0 3.5 Step 3: Use the standard normal table to find area below 1.45. Show Standard Normal Table Step 4: We conclude that P(Z>1.45)= P(Z<1.45)= 0.9265 un

(g) P(−1.10 ≤ Z ≤ 2.00)

Step 1: Sketch the curve. The probability that -1.10 Z<2.00 is equal to the blue area under the curve. 0.5 0.4 y 0.2 0.1 0.0 -1.1 2 -0.1 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Step 2: To find the probability of P (-1.10<Z<2.00). we use the following formula: P (-1.10 Z< 2.00) P(Z 2.00) P (Z<-1.10) Step 3: P (Z<2.00) can be found by using the following standard normal table. Show Standard Normal Table We see that P (Z<2.00)0.9772. = Step 4: P(Z< -1.10) can be found by using the following fomula P (Za) 1-P(Z <a) After substituting a 1.1 we have 1- P (Z < 1.1) P(Z-1.10) P(Z1.1) can be found by using the following standard normal table. Show Standard Normal Table We see that P (Z <1.1)= 0.8643 so 1- P (Z1.1) = 1-0.8643 0.1357 P(Z<-1.10) At the end we have: P (-1.10Z<2.00) - 0.8415

(h) P(1.94 ≤ Z ≤ 2.50)

Step 1: Sketch the curve Z<2,50 is equal to the blue area under the curve The probability that 1.94 0.5 0.4 0.3 y 0.2 0.1 0.0 1.94 2.5 -0.1 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 X Step 2: To find the probability of P (1.94< Z 2.50), we use the following formula: P (1.94 Z<2.50) P (Z< 2.50) P (Z < 1.94) Step 3: P (Z<2.50) can be found by using the following standard normal table. Show Standard Normal Table We see that P (Z<2.50)0.9938 Step 4: P(Z< 1.94) can be found by using the following standard normal table. Show Standard Normal Table We see that P (Z1.94) 0.9738. At the end we have: P (1.94<Z<2,50) 0,02

(i) P(1.10 ≤ Z)

Step 1: Sketch the curve. 1.10 is equal to the blue area under the curve. The probability that Z 0.5 0.4 0.3 y 0.2 0.1 1.1 -0.1 -3.5 -3.0 2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1 0 1.5 2.0 2,5 3.0 3.5 X Step 2: To find the area above 1.10, first we will find the area below 1.10 (the white one). Step 3: Use the standard normal table to find the area below 1.10. Show Standard Normal Table Step 4: We conclude that the white area 0.8643, so: Blue area 1 - white area 1 -0.8643 0.1357 Last statement can be writed in math notationa as P(Z 1.10) 1-P(Z < 1.10) = 1 0.8643 0.1357