Case Study 1 -
Edinburgh City Council is conducting a survey of people who live in the Morningside area of the city. The Council is seeking to find out the amount of money that each household spends on electricity bills every month, and offering money-saving tips to residents.
i. An interviewer will be sent out to fourteen private houses on weekday evenings over the course of one week. The City Council has previous experience in this area and believes that there is a 75% chance of finding someone at home in the evening.
a. What is the probability that there is someone at home in 11 houses?
b. What is the probability that there will be someone at home in 5 houses?
c. What is the probability that there will be someone at home in 11 or more houses?
ii. The City Council is giving away leaflets on the main street in the city centre explaining how to reduce domestic electricity bills. There is a stall that hands out the leaflets. It is known from past experience that between 12pm and 12.30pm on weekdays 10 people, on average, will stop and ask for a leaflet. Calculate the probability that on a particular weekday:
a. Exactly 8 people will stop and ask for a leaflet.
b. Between 6 and 9 people will stop and ask for a leaflet.
iii. Describe the main characteristics of distributions you have used above. Explain the importance of this distribution to management decision-making and illustrate your answer with three examples of its use in business.
Case Study 2 -
Morrbury's, a large supermarket in the UK, has two ATM machines outside its front door. The manager found that, on average, Machine 1 is used twice between 11.00 am and 11.30 am. During the same period, Machine 2 is used five times.
Required:
i. Calculate the probability that on a particular day between 11.00 am and 11.30 am:
a. Machine 1 is used 3 times.
b. Machine 1 is used at least 4 times.
c. Machine 2 is used no more than 3 times.
d. Machine 1 is used twice and Machine 2 is used once.
e. The ATMs are used at least three times.
Case Study 3-
i. A university lecturer analyzed the marks awarded for the first-year Physics exam. He found the marks to be normally distributed, with a mean of 61 and a standard deviation of 8.
a. What percentage of students scored over 73?
b. What percentage of students achieved between 51 and 65 marks?
c. If the pass mark is currently 45, what percentage of students passed the exam?
d. What percentage of students achieved less than 63 marks?
e. What is the probability of finding a student who achieved over 75 marks?
ii. The lecturer has been told that his pass rate is far too high by the university authorities. If only 80% of students are allowed to pass, then where should the lecturer set the pass mark?
iii. In the next summer's exam, students have become aware of the increased failure rate and study harder. The new average mark is 68, with the standard deviation unchanged. What mark does the lecturer have to set the pass mark to in order to still pass only 80% of students?
iv. Describe the characteristics of the normal distribution. Explain the importance of this distribution in management decision-making, giving three examples.
Case Study 4-
The Human Resources Manager at Wilson's Hotel is looking at the number of paid sick days that all employees at the hotel have been taking over the past 6 months. She has found that the average number of paid sick days taken was 6.7, with a variance of 1.44 days2.
(i) What is the probability that in a six month period an employee takes:
(a) More than 10 sick days?
(b) than 5 sick days?
(c) Between 6 and 8 days?
(ii) The Human Resources Manager has selected a sample of 40 employees at random. Assuming that the number of sick days taken has the same normal distribution as used in part (ii) above, calculate:
(a) The probability that the average number of sick days taken by the sample employees will be greater than 7 days.
(b) The probability that the average number of sick days taken by the sample employees will be less than 7?
(c) Calculate the upper and lower limits, symmetrical around 6.7 days, within which you would expect the average number of sick days taken to lie with a probability of 94%
(d) State any assumptions that are implicit in these calculations.