1 ECON20003 - QUANTITATIVE METHODS 2 Second Semester, 2020 Assignment 2 Due date and time: Monday 28 September, 4:00PM Please try to follow the marking scheme as much as you can to assure consistency across the tutorial groups. If a student makes some 'silly' mistake but otherwise is on the right track, be a bit lenient and award some mark(s) to appreciate his/her effort. Every mark should be an integer. Assignment 2 - Solutions and Answers Exercise 1 (20 marks = 12 + 8) A study of top executives' midlife crises indicates that 45% of all top executives suffer from some form of mental crisis in the years following corporate success. An executive who had undergone a midlife crisis opened a clinic providing counseling for top executives in the hope of reducing the proportion of executives who might suffer from this problem. In a random sample of 135 executives who went through this counseling program only 49 showed signs of midlife crisis eventually. (a) Given this information, use R to test at the 5% significance level whether the program is beneficial and indeed reduces the proportion of executives who show signs of midlife crisis. Specify the null and alternative hypotheses, report your observed test statistic value, make a statistical decision based on the p-value reported by R, and draw your conclusion. (12 marks: 2 marks for the correct hypotheses, 4 marks for the correct R code and printout, 2 marks for the test statistic value, 2 marks for the correct statistical decision, and 2 marks for the correct conclusion. Award at most 6 marks if the test is performed manually.) The question implies the following hypotheses: Ho : p=0.45 , HA : p<0.45 Semester 2, 2020 ECON20003 Assignment 2
2 The appropriate test is a left-tail binomial test. It can be performed by executing binom.test(49, 135, p = 0.45, alternative = "less") which returns the following printout: Exact bi nomi al test dat a: 49 and 135 number of successes = 49, number of trials = 135, p- value = 0. 02512 al ter native hypothesis: true probability of success is less than 0. 45 95 per cent confidence interval : 0. 0000000 0. 4365331 sampl e esti mat es : probability of success 0. 362963 The observed test statistic value is 49. The p-value is 0.02512 < 0.05, so at the 5% significance level we reject H0 and conclude that the program is beneficial and indeed reduces the proportion of executives who show signs of midlife crisis. (b) Are the conditions required by the test you performed likely satisfied? Explain your answer. (8 marks: 4 x 2 marks for the four assumptions with reasonable comments on them. Give at most 4 marks if there are no comments.) The exact binomial test is based on four assumptions: i. The data consists of a random sample of independent observations. The sample is said to be random. ii. The variable of interest is qualitative with two mutually exclusive but exhaustive categories. The