What is the value of the test statistic Z for testing whether the citizens of Northland and those of Southland are equally happy?

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Question 1 (21 marks) This question is concerned with using the variances of incomes from each country to compare income inequality. Is there evidence that the variances of income in the two countries are different? In your answer: (a) Specify suitable null and alternative hypotheses and define your notation. (6 marks) (b) Define a test statistic and specify its distribution assuming the null hypothesis is true. (6 marks) (c) Give the critical value(s) for a 5% significance level. (3 marks) (d) Find the value of the test statistic. (3 marks) (e) State your conclusion. (3 marks) 3 Question 2 (16 marks) In this question we investigate whether mean income is greater in Northland than it is in Southland. (a) Specify suitable null and alternative hypotheses and define your notation. (8 marks) (b) Assuming that the variances of the incomes in Northland and Southland are the same, find the p-value for a test designed to test the hypotheses in part (a). (4 marks) (c) At a 10% significance level, do you conclude that mean income in Northland is greater than mean income in Southland? (4 marks) Question 3 (16 marks) In this question we investigate the degree of poverty in each of the two countries. (a) Find estimates of the proportions of individuals who live in poverty in each of the two countries. (4 marks) (b) Find estimates of the variances of the sample proportions found in (a). (4 marks) (c) Find an estimate of the variance of the difference between the two sample proportions found in (a). (4 marks) (d) Find an estimate of the variance of the difference between the two sample proportions found in (a) assuming that the two population proportions of those living in poverty in the two countries are equal. (4 marks) Question 4 (17 marks) In this question we investigate whether the citizens of Northland and Southland are equally happy. (a) What is the value of the test statistic Z for testing whether the citizens of Northland and those of Southland are equally happy? (4 marks) (b) The test statistic in part (a) has 3 components. Let’s call them a, b and c such that a b Z c   Find a, b and c. (9 marks) (c) Would you conclude that the citizens of Northland and Southland are equally happy? (4 marks)