Question

2.1       The annual salaries of employees in a large company are approximately normally             distributed with a mean of R50 000 and a standard deviation of R 20 000.

            Calculate the percentage of employees who earn:
            2.1.1    less than R40 000?                                                                                                     
            2.1.2    between R45 000 and R65 000?                                                                                            2.1.3    more than R70,000?  

2.2       The long-distance calls made by the employees of a company are normally distributed with a mean of 6.3 minutes and a standard deviation of 2.2 minutes.

            Find the probability that a call:

2.2.1    Lasts between 5 and 10 minutes.                  

            2.2.2    Lasts more than 7 minutes.                                       

2.2.3    Lasts less than 4 minutes                                                       

                                                                                   

2.3       The probability that a driver must stop at any one traffic light coming Regent          Business          school beach Campus is 0.2. There are 15 sets of traffic lights on the journey.

            What is the probability that a student:

            2.3.1    must stop at exactly 2 of the 15 sets of traffic lights?                                               

            2.3.2    will be stopped at 1 or more of the 15 sets of traffic lights?                                      

2.4       The number of typing mistakes made by a secretary has a Poisson distribution. The         mistakes are made independently at an average rate of 1.65 per page. Find the             probability        that a three-page letter contains no mistakes.                                                               

Answer 1

P(X < A) = P(Z < (A - mean)/standard error)

2.1) Mean = 50,000

Standard deviation = 20,000

2.1.1) P(X < 40,000) = P(Z < (40,000 - 50,000)/20,000)

= P(Z < - 0.5)

= 0.3085

2.1.2) P(45,000 < X < 65,000) = P(X < 65,000) - P(Z < 45,000)

= P(Z < (65,000 - 50,000)/20,000) - P(Z < (45,000 - 50,000)/20,000)

= P(Z < 0.75) - P(Z < - 0.25)

= 0.7734 - 0.4013

= 0.3721

2.13) P(X > 70,000) = 1 - P(X < 70,000)

= 1 - P(Z < (70,000 - 50,000)/20,000)

= 1 - P(Z < 1)

= 1 - 0.8413

= 0.1587

2.2.1) P(5 < X < 10) = P(X < 10) - P(X < 5)

= P(Z < (10 - 6.3)/2.2) - P(Z < (5 - 6.3)/2.2)

= P(Z < 1.68) - P(Z < - 0.59)

= 0.9535 - 0.2776

= 0.6759

2.2.2) P(X > 7) = 1 - P(X < 7)

= 1 - P(Z < (7 - 6.3)/2.2)

= 1 - P(Z < 0.32)

= 1 - 0.6255

= 0.3745

2.2.3) P(X < 4) = P(Z < (4 - 6.3)/2.2)

= P(Z < 1.05)

= 0.8531