Question 1 (12 Marks)

The outback mining town of Two Tree Hill has no local supply of fresh water, and relies on fresh water being trucked into town. Historically the average daily consumption of fresh water is 0.632 megalitres per day with a standard deviation of 0.112 megalitres per day. They currently have 6.50 megalitres in their reservoir.

a) Assuming that water consumption follows a normal distribution, determine the probability that they consume more than 0.650 megalitres on a particular day. [2 marks]

b) If you had access to their historical water consumption records, suggest two ways that you could quickly check if this normality assumption appears reasonable.[2 marks]

c) Their next water delivery is scheduled to occur in 10 days' time. Determine the probability that they will run out of water if they follow their historical consumption patterns. (That is, determine the probability that average daily consumption exceeds 0.65 megalitres over the next 10 days.) [4 marks]

The Town Council at Two Tree Hill is considering building a bigger reservoir to reduce their exposure to water shortages. However the cost of this project would mean that they would have to increase the rates that Two Tree Hill's 2280 citizens have to pay, and it is not obvious whether the citizens would be in favour of such a capital project. The council arranges for a random sample of 400 citizens to be surveyed, and this survey reveals that 214 (53.5%) are in favour of the project.

d) Determine a 95% confidence interval for the proportion of ALL of the town's citizens that are in favour of the project. [4 marks]