Turtle’s limousine service has determined that the number of kilometres each limousine travels each year

INSTRUCTIONS TO CANDIDATES

ANSWER ALL QUESTIONS

Question 1 [10 marks: 2 each]

Turtle’s limousine service has determined that the number of kilometres each limousine travels each year is normally distributed with a mean of 100,000 kilometres and a standard deviation of 12,000 kilometres.

a)            What proportion of these limousines can be expected to travel more than 120,000 kilometres in the next year?

b)            What proportion of these limousines can be expected to travel at least 120,000 kilometres in the next year?

c)            How many kilometres will be travelled by at least 90% of the limousines in the next year?

d)            How many kilometres will 98% of the limousines do less than in the next year?

e)            85% of the limousines will do between what two distances (in kilometres), given that the two amounts must be the same distance from the mean?

Question 2 [7 marks: 4 for a) and 3 for b)]

As the production manager of a continuous A4 paper production line, you sample the length of the paper at (fixed) periodic intervals. A4 paper should be 297mm long and the machines used claim a standard deviation of 2mm. You plan to take a sample of 100 sheets to test paper length.

a)            What is the probability that the sample mean will be between 296.5mm and 297.4mm?

b)            What type of sampling procedure has been used and is it likely to have created a biased sample? Briefly discuss.

Question 3 [8 marks: 2 each] 

A continuous random variable Y has a probability density function f(y) = 1 24 ,  0 ≤ y ≤ 24. 

a)            What is the mean and standard deviation of Y?

b)            What is the probability that Y is exactly 6?

c)            What is the probability that Y is less than 10?

d)            Briefly describe a real-world example that could be modelled by Y. 

Question 4 (continuation of Question from HW2) [6 marks: 2 each]

Recall that customers arrive at Flourish and Blotts randomly and independently of each other at a rate of 10 per day, on average. Assume that Flourish and Blotts is open 24 hours a day and a customer has just arrived.

a)            What is the average time between customers and the standard deviation of the time between customers?

b)            What is the probability that no customers arrive in the next hour?

c)            What is the probability that the next customer arrives in 90 – 120 minutes time?

Question 5 (continuation of Question from HW1) [6 marks]

Ann’s Facebook page has 170,750 Likes. A survey was posted on Ann’s Facebook page about whether people who Liked the page would by an ebook authored by her. Assume that the truth is 5% would buy.

a)            What is the probability that a survey of 250 respondents would indicate 5% or more would buy an ebook?                [4 marks]

b)            Is it likely that the sampling procedure created a biased sample? Discuss.         [2 marks]

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