Question 2

1. A random sample of 270 customers of an online chat service were surveyed. They were asked a question about the likelihood of them recommending the service to others, and this information was used to classify their attitude as a Detractor, Passive or Advocate. They were also asked how they expect their use of the chat service to change over the coming year. The data is summarised in the contingency table below. (Unfortunately, some of the data is missing from the table.)

	Detractor	Passive	Advocate	TOTAL
Increase	10			71
Stay the same		60	34
Decrease	30	46
TOTAL	55		88	270

Required

1. Fill in the missing numbers.

	Detractor	Passive	Advocate	TOTAL
Increase	10	21 (71-(10+40)	40 (88-(14+34)	71
Stay the same	15 (55-(10+30)	60	34	109 (15+60+34)
Decrease	30	46	14 (90-(30+46)	90 (270-(71+109)
TOTAL	55	127 (20+60+46)	88	270

[2 marks]

1. Determine the probability that a randomly selected customer expects to increase their use of the service over the coming year.

Total customers who increase the use of service (E₁) = 71

Total customers = 270

P(E) = 71/270 = 0.263

[1 mark]

1. A randomly selected customer is an "advocate". What is the probability that they expect to increase their use of the service over the coming year?

[2 marks]

Total advocate = 88

No. of an advocate who increases service use (E₁) = 40

P(E₁) = 40/88 = 0.4545

1. Explain whether attitude and expected usage are independent.

[1 mark]

1. Over a long period of time, an electrician has established that 20% of his accounts are paid late. A friend suggests that changing the ink colour on his accounts from black to red might help encourage customers to pay more quickly. To test this idea, the electrician sends accounts printed in red ink to a random sample of 40 of his customers. Only one of these customers is late in paying their account.

Required

1. Using an appropriate probability distribution, determine the probability of having no more than one late payment from 40 customers, if the late payment rate is 20%.

[4 marks]

1. Explain whether this probability provides strong evidence that using red ink has helped reduce the late payment rate.

[2 marks]

1. A company produces bottles of mineral water, which are labelled as having 1000ml contents. The bottling machine is old, and the amount of water varies from bottle to bottle. Records show that the amount of water in a bottle has a mean of 1060 ml and a standard deviation of 14 ml, when it is running correctly.

Required

1. A bottle is randomly selected and found to contain 998 ml of water. Determine the probabilityof a bottle containing no more than 998 ml, noting all relevant assumptions. Explain whether this probability is suggestive of the fact that the bottling machine is running correctly.

[3+1=4 marks]

1. Five bottles are randomly selected and found to contain and average of 999 ml of water. Determine the probability of five bottles containing an average of no more than 999 ml. Explain whether this probability suggests that the bottling machine is running correctly or not.

[4+1 = 5 marks]

1. The diameter of gears produced by a milling machine follows a normal distribution with a mean of 25.045 mm and a standard deviation of 0.018 mm. If a gear has a diameter of less than 25.00 mm it is too small and needs to be scrapped. If a gear has a diameter more than 25.05 mm it is too large and needs to be scrapped.

Required

What proportion of gears are the right size (that is, don't need to be scrapped)?

[3 marks]

[Question 2 = 24 marks]