MGMT2262 Worksheet #4 Solutions

 

Question 1

a)      P(x < 3.9) = (3.9 – 3.85)/(4.15 – 3.85) = 0.05/0.3 = 0.1667 = 16.67%

b)     P(x > 4.05) = (4.15 – 4.05)/(4.15 – 3.85) = 0.1/0.3 = 0.3333 = 33.33%

c)      m = (3.85 + 4.15)/2 = 4; s = (4.15 – 3.85)/sqrt(12) = 0.3/sqrt(12) = 0.0866

m - 0.5s = 4 – 0.5(0.0866) = 4 – 0.0433 = 3.9567

m + 0.5s = 4 + 0.0433 = 4.0433

P(3.9567 < x < 4.0433) = (4.0433 – 3.9567)/(4.15 – 3.85) = 0.0866/0.3 = 0.2887 = 28.87%

This does not seem like a realistic expectation.

 

Question 2

a)      2 customers per minute = 1 customer per 0.5 minute; m = 0.5; P(X > 2) = e-2/0.5 = e-4 = 0.0183 = 1.83%

b)     P(X < 1) = 1 – e-1/0.5 = 1 – e-2 = 1- 0.1353 = 0.8647 = 86.47%

 

Question 3

a)      m = 10: P(X > 10) = e-10/10 = e-1 = 0.3679 = 36.79%

b)     m = 10 x 6/3 = 20; P(X > 10) = e-10/20 = e-0.5 = 0.6065 = 60.65%

c)      m = 8 x 6/4 = 12; P(X > 15 | X > 10) = P(X > 5) = e-5/12 = 0.6592 = 65.92%

 

Question 4

a)      0.4429 from the table

b)     0.4993 from the table

c)      0.3997 + 0.2486 = 0.6483

d)     0.4945 – 0.4177 = 0.0768

e)      0.4761 – 0.008 = 0.4681

f)       We need the 85.08th percentile. From this we get P(0 < Z < x) = 0.3508. Looking for the probability 0.3508 in the Z table, we find the corresponding Z value to be 1.04

g)      Since P(-x < Z < x) = 0.9756, P(0 < Z < x) = 0.4878. As with part f, we find the corresponding Z value to be 2.25

 

Question 5

a)      Z = (7-5)/2 = 1. Then P(X < 7) = P(Z < 1) = 0.5 + 0.3413 = 0.8413.

b)     Z = (9.2-5)/2 = 2.1. Then P(X > 9.2) = P(Z > 2.1) = 0.5 – 0.4821 = 0.0179

c)      Z1 = (4-5)/2 = -0.5; Z2 = (6.4-5)/2 = 0.7. Then P(4 < X < 6.4) = P(-0.5 < Z < 0.7) = 0.1915 + 0.258 = 0.4495

d)     Since, we need the 93.7th percentile, we get P(0 < Z < x) = 0.437. The Z value is 1.53. Then 1.53 = (x-5)/2. Solving for x, x = 5 + 2(1.53) = 8.06.

e)      Using the empirical rule for symmetric data, m - 2s = 5 – 2(2) = 1; m + 2s = 5 + 2(2) = 9

 

Question 6

a)      Z = (750-800)/75 = -0.67. Then P(X < 750) = P(Z < -0.67) = 0.5 – 0.2486 = 0.2514

b)     Z = (900-800)/75 = 1.33. Then P(X > 900) = P(Z > 1.33) = 0.5 – 0.4082 = 0.0918

c)      We need the 95th percentile which is 1.645. Then 1.645 = (X – 800)/75 from which we get X = 800 + 1.645(75) = 923.38.

d)     Let x be the upper Z value. Then P(-x < Z < x) = 0.796 from which we get P(0 < Z < x) = 0.398. The upper Z value is 1.27 and the lower Z value is -1.27. Solving for the upper value, we have 1.27 = (X – 800)/75 from which we get X = 800 + 1.27(75) = 895.25. The lower value is 800 – 1.27(75) = 704.75.

 

Question 7

a)      Z = (10.01-10.2)/0.2 = -0.95. Then P(X < 10.01) = P(Z < -0.95) = 0.5 – 0.3289 = 0.1711

b)     Z = (10.3-10.2)/0.2 = 0.5. Then P(X > 10.3) = P(Z > 0.5) = 0.5 – 0.1915 = 0.3085

c)      In this case we need the 5th percentile of Z which is –1.645. Then –1.645 = (x-10.2)/0.2. Solving for x, x = 10.2 – 1.645(0.2) = 9.871 seconds.

d)     Using the empirical rule for symmetric data, m - 3s = 10.2 – 3(0.2) = 9.6; m + 3s = 10.2 + 3(0.2) = 10.8