09/06/2022, 00:21 Quiz: Final 2021SM1 with solution https://canvas.lms.unimelb.edu.au/courses/124333/quizzes/157982/take 1/10 Final 2021SM1 with solution Started: Jun 9 at 2:19 Quiz Instructions 18 ptsQuestion 1 A newly opened driving school is running a promotion to advertise their lessons to new drivers with learner permits. For each of the next 120 new drivers who receive their learner permit, a representative of the driving school invites them to flip a coin : “heads” means the new driver can take driving lessons for free, “tails” means the new driver will not receive any driving lessons. Let N be a random variable denoting the number of new drivers who take the driving lessons. 1. (3 marks) What is the probability distribution of N 2. (2 marks) What is the expected value of N The probability that a new driver will progress to passing their driving test without driving lessons is 0.8, while those who have taken the lessons would pass their driving test with probability 0.9. Let P be a random variable denoting the number out of all of the new drivers with learner permits who pass their driving test. 3. (6 marks) Derive an expression for E( P | N ). (Hint: it may help to write P = P + P , where P is the number who passed the driving test out of those who flipped “heads” (and hence received the lessons), and P is the number who passed the driving test out of those who flipped “tails” (and missed out on the lessons). 4. (3 marks) Hence calculate E( P ). 5. (4 marks) After the promotion is completed you meet a new driver who has passed their driving test. What is the probability that they took the driving lessons H T H T Edit View Insert Format Tools Table 12pt Paragraph 09/06/2022, 00:21 Quiz: Final 2021SM1 with solution https://canvas.lms.unimelb.edu.au/courses/124333/quizzes/157982/take 2/10 p 0 words Ginger is now ac Start writing! Your mistakes wi 10 ptsQuestion 2 There is growing international interest is comparing hospitalisation rates (proportions of patients entering hospital) for older male and female patients with Covid-19. In particular there is a hypothesis that the male hospitalisation rate is higher than the female hospitalisation rate. In order to carry out a study on this, independent simple random samples were obtained of 80 male and 80 female patients over the age of 70 who tested positive to Covid-19. Each of these patients was monitored until they either recovered at home or were hospitalised, and the numbers of each for males and females recorded. Define the notation p : the population hospitalisation rate for older male patients with Covid-19 p : the population hospitalisation rate for older female patients with Covid-19 (i) To test the theory, the null hypothesis would be a. H : p = p b. H : p > p c. H : p < p d. H : p ≠ p (ii) To test the theory, the alternative hypothesis would be a. H : p = p b. H : p > p c. H : p < p d. H : p ≠ p M F 0 M F 0 M F 0 M F 0 M F A M F A M F A M F A M F 09/06/2022, 00:21 Quiz: Final 2021SM1 with solution https://canvas.lms.unimelb.edu.au/courses/124333/quizzes/157982/take 3/10 (iii) Suppose it is genuinely true that hospitalisation rates are higher for males than for females. If we carried out a hypothesis test and did not reject the null hypothesis at the 5% level of significance, this would be: a. a correct decision b. a Type I error c. a Type II error (iv) Suppose it is genuinely true that hospitalisation rates are equal for males and females. If we carried out a hypothesis test and did not reject the null hypothesis at the 5% level of significance, this would be: a. a correct decision b. a Type I error c. a Type II error (v) An independent statistician evaluating the study plan is concerned the sample size is not large enough to reliably detect the hypothesised difference in hospitalisation rates if it is genuinely present. A larger sample would address this concern because it would a. improve the significance level of the hypothesis test b. improve the power of the hypothesis test c. reduce the probability of a Type I error d. imply that the Central Limit Theorem is not necessary e. none of these (vi) The sample design for the study can be best described as a. matched pairs of male and female patients b. a single simple random sample of 160 patients c. independent samples of male and female patients d. all of these (vii) When the study was carried out, there were 14 recorded hospitalisations amongst the male patients and 10 amongst the females. The sample hospitalisation rates are therefore a. p = 0.175, p = 0.125 b. p = 0.125, p = 0.175 c. p = 0.175, p = 0.125 d. p = 0.125, p = 0.175 (viii) The standard error of the difference between the male and female sample hospitalisation rates is closest to a. 0.042 b. 0.056 c. 0.037 d. 0.888 M F M F M F M F 09/06/2022, 00:21 Quiz: Final 2021SM1 with solution https://canvas.lms.unimelb.edu.au/courses/124333/quizzes/157982/take 4/10 p 0 words (ix) The p value for the hypothesis test is closest to a. 0.376 b. 0.05 c. 0.188 d. 0.812 (x) At the 5% level of significance, the conclusion of this test is 1. Reject H and conclude that there is evidence that the male hospitalisation rate is higher than the female rate 2. Reject H and conclude that there is no evidence that the male hospitalisation rate is higher than the female rate 3. Do not reject H and conclude that there is evidence that the male hospitalisation rate is higher than the female rate 4. Do not reject H and conclude that there is no evidence that the male hospitalisation rate is higher than the female rate 0 0 0 0 Looks like you got si In order to continue receiving th Ginger, please sign b SIGN BACK IN Don’t show this ag 12 ptsQuestion 3 The 24/7 chain of small convenience stores has been having problems with shoplifting, so they installed some CCTV cameras in visible places in each of their Edit View Insert Format Tools Table 12pt Paragraph 09/06/2022, 00:21 Quiz: Final 2021SM1 with solution https://canvas.lms.unimelb.edu.au/courses/124333/quizzes/157982/take 5/10 p 0 words stores to try to deter offenders. In order to assess the effect of these cameras, they chose a random sample of 35 of their stores and gathered some data on the value of thefts over a four week period prior to the installation of the cameras. For these stores the mean of the losses to theft was $1,336 with standard deviation of $437. Data were then collected for the value of thefts in the same stores over the four week period following the installation of the cameras. The mean losses were $1,299 with standard deviation of $312. The standard deviation of the changes in losses for each store was $266. 1. Carry out a hypothesis test to evaluate whether the presence of the CCTV cameras made any difference to the mean losses because of theft. Use the 5% level of significance and a p value decision rule. Include (a) statement of hypotheses (including definition of any notation) (b) brief justification of testing approach chosen (c) test statistics and decision rule (d) decision of the test and brief interpretation 2. Compute a 90% confidence interval for the change in the mean losses to theft as a result of the CCTV installation. Looks like you got si In order to continue receiving th Ginger, please sign b SIGN BACK IN Don’t show this ag 12 ptsQuestion 4 Edit View Insert Format Tools Table 12pt Paragraph 09/06/2022, 00:21 Quiz: Final 2021SM1 with solution https://canvas.lms.unimelb.edu.au/courses/124333/quizzes/157982/take 6/10 To evaluate the statistical relationship between the profitability of firms and the educational qualifications of their CEOs, a simple random sample of 119 firms was collected and data on the following two variables obtained: Profit = Annual profit of firm i in millions of dollars Grad = 1 if the CEO of firm i had a post-graduate degree, = 0 otherwise. The sample mean of the observations for Profit was calculated to be 0.238 ($million), with standard deviation of 0.455. There were 68 CEOs in the sample who had a post-graduate degree. The sample covariance between Profit and Grad was 0.013. 1. (1 mark) What is the sample mean of Grad 2. (2 marks) The sample variance of Grad was calculated to be 0.247. Briefly explain how this was calculated using only the information given in this question. 3. (3 marks) Calculate a 95% confidence interval for the proportion of firms whose CEOs have a post-graduate degree. (Report your standard error and critical value as well as the confidence interval itself.) 4. (2 marks) Consider a regression of the form E ( Profit | Grad ) = β + β Grad Use the provided sample statistics to calculate β and β . 5. (2 marks) Use your regression results to calculate the sample mean of profits for firms whose CEOs do not have a post-graduate degrees. Explain how you did the calculation. 6. (2 marks) Use your regression results to calculate the sample mean of profits for firms whose CEOs do have a post-graduate degree. Explain how you did the calculation. i i i i i i i i i 0 1 i 0 1 Looks like you got si In order to continue receiving th Ginger, please sign b SIGN BACK IN Don’t show this ag Edit View Insert Format Tools Table 12pt Paragraph 09/06/2022, 00:21 Quiz: Final 2021SM1 with solution https://canvas.lms.unimelb.edu.au/courses/124333/quizzes/157982/take 7/10 p 0 words 15 ptsQuestion 5 As an analyst of firms characteristics and profitability, you are considering the relationship between firm profitability and the amount paid to their CEOs. A simple random sample of 117 firms was collected and data on the following two variables obtained: Profit = Annual profit of firm i in millions of dollars Salary = Salary of the CEO in millions of dollars A regression of Profit on Salary is estimated, given the following equation, (with standard errors in brackets): ê(Profit |Salary ) = -0.197 + 0.33 Salary (0.071) (0.048) Use these results to answer the following questions. 1. (2 marks) Give an interpretation of the coefficient on Salary 2. (2 marks) Construct a 95% confidence interval for the population coefficient on Salary in this equation. Give the critical value as well as the confidence interval. 3. (3 marks) Your boss claims there should be some statistical relationship between firm profitability and CEO salary. Use your previous answer to evaluate this claim. 4. (3 marks) Your boss further claims that paying a CEO a higher salary will cause the CEO to perform better and hence produce higher firm profitability. Is this claim confirmed by the regression results Explain. 5. (2 marks) The R for the regression is 0.289. What does this measure and what are its implications for this regression 6. (3 marks) Consider the scatter plot below of Profit and Salary . What do you observe about this plot that might be a problem for statistical inference in the regression Explain. i i i i i i i i i 2 i i 09/06/2022, 00:21 Quiz: Final 2021SM1 with solution https://canvas.lms.unimelb.edu.au/courses/124333/quizzes/157982/take 8/10 p 0 words 5 ptsQuestion 6 Edit View Insert Format Tools Table 12pt Paragraph 09/06/2022, 00:21 Quiz: Final 2021SM1 with solution https://canvas.lms.unimelb.edu.au/courses/124333/quizzes/157982/take 9/10 The diagram below presents per capita GDP (in 2019 in U.S dollars) and obesity rate (percentage of population classified as obese) among 154 countries in 2019. In no more than 200 words, write a description of what this diagram is and what it reveals. END OF EXAM Press ‘Submit Quiz’ button to finish. 09/06/2022, 00:21 Quiz: Final 2021SM1 with solution https://canvas.lms.unimelb.edu.au/courses/124333/quizzes/157982/take 10/10 No new data to save. 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