This question is inspired by the report ‘A majority of First Nations peo-ple support the voice. Why don’t non-Indigenous Australians believe this?’,Rebecca Huntley (2023)The Guardian Australia (https://www.theguardian.com/commentisfree/2023/apr/27/a-majority-of-first-nations-people-support-the-voice-why-dont-non-indigenous-australians-believe-this), also linked onthe LMS.You DO NOT need information from this article to answer thequestions; it is provided for context only.This report discusses the results from a poll of 738 Indigenous people,where they were asked whether or not they support including an Indigenousvoice to parliament in the Australian constitution. The referendum (elec-tion) will be held on 14th October 2023. The poll identified 613 Indigenouspeople who support the inclusion of the voice in the constitution.(a). Construct a 95% confidence interval for the proportion of IndigenousAustralians who support the voice to parliament being included in theconstitution.(b). The official case for the ‘No’ campaign for the referendum (availablehere: https://www.aec.gov.au/referendums/learn/your-official-referendum-booklet.html, and on the LMS) claims that “[m]any Indigenous Aus-tralians do not support [the voice].”Would it be reasonable to state that “most Indigenous Australiansdo not support the voice”? Justify your answer with appropriate evi-dence.(c). Outline how you would collect a representative sample of non-IndigenousAustralian adults, of a similar size to this poll. Your answer will beassessed both on the statistical validity and the practicality of yourdesign.(d). A sample of 100 non-Indigenous voters in Victoria found that manywere still undecided. Of the 59 people who had decided how they weregoing to vote, 39 supported an Indigenous voice to parliament in theconstitution. Conduct an approximate hypothesis test to determine ifthis sample provides evidence a majority of the Victorian populationsupport the voice. Show all of your calculations and steps.Your answer needs to (the 5 step process meets these requirements):ˆ State the hypotheses in terms of the parameter(s) of interest.ˆ Calculate sd(estimator).ˆ Calculate the test statistic, and give its distribution under thenull hypothesis.ˆ Give the P -value for the test, using Minitab (you should not useMinitab for other parts of this question).ˆ State your conclusion in the context of the data.(e). It is believed that the final result of the vote may be very close to50%. Researchers would like to estimate this proportion using an 80%confidence interval based on a normal approximation, with a maximummargin of error of 0.02. What sample size would be required to achievethis? Show your calculations as well as your answer.

(a). To construct a 95% confidence interval for the proportion of Indigenous Australians who support the voice to parliament being included in the constitution, we first need to calculate the sample proportion (p̂) and the standard error (SE).

The sample proportion (p̂) is calculated as the number of successes (in this case, the number of Indigenous people who support the voice) divided by the total number of trials (the total number of Indigenous people polled). So, p̂ = 613 / 738 = 0.831.

The standard error (SE) for a proportion is calculated as sqrt[(p̂ * (1 - p̂)) / n], where n is the total number of trials. So, SE = sqrt[(0.831 * (1 - 0.831)) / 738] = 0.015.

A 95% confidence interval is calculated as p̂ ± (1.96 * SE). So, the 95% confidence interval is 0.831 ± (1.96 * 0.015) = (0.801, 0.861).

(b). Based on the poll results, it would not be reasonable to state that "most Indigenous Australians do not support the voice". The poll identified that a majority (83.1%) of the polled Indigenous people support the inclusion of the voice in the constitution. This is strong evidence against the claim that most Indigenous Australians do not support the voice.

(c). To collect a representative sample of non-Indigenous Australian adults, a stratified random sampling method could be used. This involves dividing the population into different subgroups (or strata) based on relevant characteristics (such as age, gender, location, etc.), and then randomly selecting individuals from each subgroup. This ensures that the sample is representative of the population in terms of these characteristics. The sample size from each stratum could be proportional to the size of the stratum in the population to ensure representativeness.

(d). To conduct a hypothesis test, we first state the null hypothesis (H0) and the alternative hypothesis (H1). H0: p = 0.5 (the proportion of non-Indigenous voters in Victoria who support the voice is 50%), H1: p > 0.5 (the proportion is greater than 50%).

The sample proportion (p̂) is 39 / 59 = 0.661. The standard error (SE) is sqrt[(0.5 * (1 - 0.5)) / 59] = 0.065.

The test statistic (Z) is (p̂ - p) / SE = (0.661 - 0.5) / 0.065 = 2.48. Under the null hypothesis, this follows a standard normal distribution.

The P-value is the probability of observing a test statistic as extreme as, or more extreme than, the observed test statistic, under the null hypothesis. This can be calculated using Minitab or a standard normal distribution table. If the P-value is less than the significance level (typically 0.05), we reject the null hypothesis.

(e). To estimate the proportion with a maximum margin of error of 0.02 using an 80% confidence interval, we first need to find the z-score corresponding to an 80% confidence level, which is 1.28.

The sample size (n) is calculated as (Z^2 * p * (1 - p)) / E^2, where E is the margin of error. Assuming p = 0.5 (as we don't know the true proportion), n = (1.28^2 * 0.5 * (1 - 0.5)) / 0.02^2 = 2560. So, a sample size of 2560 would be required.