Question 2

(a) 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

(i) Fill in the missing numbers.

[2 marks]

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

[1 mark]

(iii) 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]

(iv) Explain whether attitude and expected usage are independent.

[1 mark]

(b) Over a long period of time, an electrician has established that 17% 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 30 of his customers. Only one of these customers is late in paying their account.

Required

(i) Using an appropriate probability distribution, determine the probability of having no more than one late payment from 30 customers, if the late payment rate is 17%.

[4 marks]

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

[2 marks]

(c) 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 1015 ml and a standard deviation of 10 ml, when it is running correctly.

Required

(i) A bottle is randomly selected and found to contain 999 ml of water. Determine the probability of a bottle containing no more than 999 ml, noting any assumptions that you need to make. Explain whether this probability suggests that the bottling machine is running correctly.

[3+1=4 marks]

(ii) 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.

[4+1 = 5 marks]

(d) The diameter of gears produced by a milling machine follows a normal distribution with a mean of 25.03 mm and a standard deviation of 0.012 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]