Question: This assignment covers material in chapters 10 to 15 of the Unit Notes. R must be used for all graphs and statistical analyses based on raw data. Attach relevant R output to support your answers. All data files mentioned are available from the Data Files page on LMS.

1. a) Public health authorities want to estimate the proportion of Australians aged 18-65 who caught the flu during winter 2016. If the true proportion is thought to be about 23%, how large a sample will be needed if we want to be 95% sure that our estimate will be within 0.03 of the true value?

(b) Due to a change in the available funding, the sample actually used for estimating the prevalence of flu included 2,553 randomly-selected Australians aged 18-65. Of these, 615 reported having a "flu-like illness" during winter 2016. Use these data to calculate a 90% confidence interval for the proportion of Australians aged 18-65 who had a flu-like illness during winter 2016.

2. Fresh water from rivers and lakes varies in acidity/alkalinity depending on the rocks and soils it runs over. Granite country yields water that is slightly acidic, while water from limestone country is slightly alkaline. Acidity and alkalinity are measured on a pH scale, on which 7 is
neutral, values below 7 are acidic and values above 7 are alkaline. pH can change due to pollution or due to waste from biomass such as algae.

A certain river, fed by various tributaries, usually has a pH of about 6.3 (slightly acidic). The file Water_pH.csv contains pH values from randomly-selected sites on this river and its tributaries.

(a) Provide a graph that displays the distribution of the data. Bear in mind that distribution shape and outliers may be important. [5]

(b) How much evidence is there that the pH of water in this river system has changed? Evaluate the evidence using a hypothesis test with a significance level of 1%. Take care to state and evaluate all assumptions, and provide an appropriate conclusion. [12]

(c) Provide a 99% confidence interval for the mean pH in this river system. [3]

3. A double-blind randomised experiment investigated whether consumption of oligofructose would stimulate calcium absorption in healthy human males aged 14 to 16 years. Two sets of pills were prepared, identical in appearance and packaging. One set contained oligofructose
and the other set contained ordinary sucrose (i.e., a placebo). The experiment ran in two rounds, with the second round starting 3 weeks after the first ended. Subjects were randomly assigned to either receive oligofructose in the first round and the placebo in the second, or the placebo in the first round and oligofructose in the second. In each round, subjects took pills for nine days and had their calcium absorption measured on the last day. Calcium absorption was measured as a percentage of calcium intake. The 3 week period between rounds ensured that the sugars (oligofructose and sucrose) had been completely excreted before the second round commenced.

The calcium absorption data are available in two data files, Oligo_paired.csv and Oligo_indep.csv. Only one of these is appropriate to the problem, though the same calcium absorption measurements appear in each file.

In Oligo_paired.csv, the data columns are headed "Control" and "Oligofructose" to indicate which pill was being taken when calcium absorption was measured. Each row contains two calcium absorption measurements from a single subject.

In Oligo_indep.csv, the columns are headed "CaAbsorb" and "Pill". CaAbsorb contains calcium absorption measurements, and "Pill" indicates whether the pill being taken before each measurement was a placebo ("P") or oligofructose ("O").

(a) [Note: this part is worth 4 marks because it is important to the rest of the question, not because it needs a long answer. A correct answer can be quite short. Follow-through marks will be available if you get this part wrong.]

Should the data be analysed as independent samples, or as paired data? Justify your answer.

(b) Provide an appropriate display of the data distribution, bearing in mind your choice of analysis.

(c) Formally test whether oligofructose stimulates calcium absorption in healthy males aged 14 to 16 years. You do not need to revisit the decision made in part (a), but your report should be consistent with that decision, and should include all relevant elements of a hypothesis
test. Assume α = .05.