Answer the following questions with working step

Q1. 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.

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.

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.)

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.

Q2. The Department of Consumer Affairs is interested in difference, D, between the price that people are quoted online for a taxi trip, and the price that they are actually charged. They seek online quotes for cost of a taxi trip between twelve pairs of random locations, and then take these taxi rides and record the actual price changed. They subtract the quoted price from the charged price. The results (in $), and their accumulations appear below.

25.10

12.45

-2.60

5.60

13.80

-1.20

7.30

0.10

15.10

3.15

9.95

17.35

a) Determine the mean and standard deviation of price differences for this sample.

b) The Taxi Owners' Association claims that "the average difference between charged and quoted prices is no more than $5.00". Using = 5% and stating any assumptions required, test the Association's claim.

c) Explain the difference between statistical significance and practical significance, and state whether your mean difference between quoted and actual price is statistically significant. How would you interpret an outcome that was statistically significant but not practically significant?

d) Using the sample standard deviation from part (a) as an estimate for population standard deviation, determine the sample size required in order to be 90% confident of estimating mean price difference within $1.00 of its true value.

Q3. Each of following summaries of hypothesis tests contain one or more errors (possibly including calculation errors) or issues that invalidate the process or the conclusion. Identify and describe these.

a) Scenario: Mean hourly production was historically 101.1 kg per hour with a standard deviation of 23.4 kg. After a process change a random sample of 87 hours reveals a sample mean of 107.4 kg per hour. Has average hourly production changed

Ho: u = 43.2 vs HA:u # 43.2 Y = 41.9, s =10.2 , n=13, a=0.10, N very large and unknown Ztest = -0.460. Zcrit = +/-1.645 Decision: retain Ho Conclusion: there is less average protein than they are claiming.Ho: u = 65 vs HA: + 65 X = 59.5, s =18.1 , n=25, d=0.05, N = 150, sampling without replacement test = -1.9218, terit = +/-2.0227 Decision: retain Ho Conclusion: the data is consistent with achieving an overall average mark of 65.Ho:u = 101.1 vs HA: p # 101.1 X = 107.4, o=23.4, n=87, o=0.05, N very large and unknown Ztest = 2.511, Zedit= +/-1.96 Decision: retain Ho Conclusion: Average production hasn't changed.Ho:p = 0.5 (alternatively expressed as Ho:p 2 0.5 ) vs HA:p