A - Short Questions Instructions: Answer all questions in your exam booklet. 1. A module has three assessment components: an initial assessment with a 20% weighting, a project with a 30% weighting and a final exam with a 50% weighting. Assuming every assessment has a full mark of 100 marks, calculate the final mark of a student who scored 65 marks in his initial assessment, 70 marks for his project and 50 marks for his final exam. 2. Given a normal distribution with = 50 and = 4, what is the probability that : a) X > 43. b) 5% of the values are less than what X value? c) 60% of the values are between what two X values (symmetrically distributed around the mean)? 3. If the sample mean is 50 kg, the sample standard deviation is 15 kg and assuming that the population is normally distributed, construct a 99% confidence interval estimate of the population mean if the sample size of 16 is selected. 4. Determine the critical value of the student's t-distribution for a two-tailed hypothesis test at a 0.10 level of significance based on a sample of size 24. 5. The households in Lonsville town were surveyed to determine whether they would subscribe to a new Pay TV channel. The households were classified according to \"high', \"medium' and 'low' income levels. The results of the survey are summarized in the table below Income Level High Medium Low Will subscribe 3,200 1,920 480 Will not subscribe 800 7,080 2,520 a) What is the marginal probability that a household will subscribe? b) What is the joint probability that a household will subscribe and be high income? c) What is the probability that a high-income household will subscribe? Page 1 of 5 d) Is income level statistically independent of whether a household subscribes or not? Explain. 6. A new drug is found to be effective on 90% of the patients tested. If the drug is administered to 20 randomly chosen patients at a large hospital, find the probability that it is effective on: a) Fewer than five of the patients. b) 10 or more of the patients. c) All 20 of the patients. 7. The number of floods in a certain region is approximately Poisson distributed with an average of three floods every 10 years. Find the probability that a family living in the area for one year will have experienced: a) Exactly one flood. b) At least one flood. c) Find the probability that a student who moves to the area for three years will experience exactly one flood. 8. Briefly describe the two types of error that can arise from hypothesis testing, and explain the relationship between the two types of error. 9. An ATM must be stocked with enough cash to satisfy customers making withdrawals over an entire weekend. If too much cash is kept unnecessarily in the ATMs, the bank is forgoing the opportunity of investing the money and earning interest. At a particular branch, the population mean amount of money withdrawn from the ATM per customer transaction over the weekend is $220 with a population standard deviation of $30. A random sample of 36 customer transactions indicates that the sample mean withdrawal amount is $232. Is there evidence to believe that the population mean withdrawal amount is no longer $220? (Use a 0.05 level of significance.) 10. Describe in at most three sentences what is meant by the term \"correlation coefficient\". Section B - Long Questions Page 2 of 5 Instructions: Answer all 3 questions. Detailed answers are required to obtain full marks. 1. A bank branch located in a commercial district of a city has developed an improved process for serving customers during the noon to 1pm lunch period. The waiting time in minutes for all customers during this hour is recorded over a period of one week. A random sample of 15 customers is selected. And the results are as follows: 4.21 5.12 a) 5.55 6.46 3.02 6.19 5.13 3.79 4.77 2.34 3.54 3.20 4.50 6.10 0.38 Calculate the mean, mode and median. [3 marks] b) Calculate the variance, standard deviation, interquartile range, coefficient of variation and the Z-score of the 2 extreme numbers. Are there any outliers? Explain. [9 marks] c) Are the data skewed? Explain. [3 marks] d) As a customer walks into the branch office during the lunch hour, she asked the manager how long she can expect to wait. The manager replies, \"Almost certainly less than five minutes\". On the basis of the results of (a) and (b) above, evaluate the accuracy of this statement. [5 marks] Page 3 of 5 2. a) Give a short explanation of Bayes' Theorem; write at most three sentences; you may use diagrams to illustrate your answer. [3 marks] b) The probability that a person has a certain disease is 0.03. Medical diagnostic tests are available to determine whether a person has the disease. If the disease is present, the probability that the medical diagnostic test will give a positive result (indicating that the disease is present) is 0.90. If the disease is not present, the probability of a positive test result is 0.002. Suppose that the medical diagnostic test has given a positive result. What is the probability that the disease is present, given that positive test result? [5 marks] c) The time between unplanned shutdowns of a power plant has an exponential distribution with a mean of 20 days. Find the probability that the time between two unplanned shutdowns is (i) Less than 14 days [2 marks] (ii) More than 21 days [2 marks] (iii) Less than 7 days. [2 marks] d) What is the meaning of Mutually Exclusive and Collectively Exhaustive? e) The market research manager at Shesheido Beauty Pte. Ltd. wants to study women's spending on cosmetics. A survey is designed to estimate the proportion of women who purchase their cosmetics primarily from Shesheido, and the mean yearly amount that women spent on cosmetics. A previous survey found that the standard deviation of the amount women spend on cosmetics in a year is approximately $18. Determine the sample size needed to have a 99% confidence of estimating the population mean to within $5. [4 marks] Page 4 of 5 [2 marks] 3. An increasingly important topic in the context of climate change and global warming is the link between countries' energy generation and CO 2 emissions. The following table lists energy generation (terawatt hours) and CO2 emissions ('000s metric tonnes) for 11 countries. Country a) CO2 Energy USA 5,762,050 4,150 China 3,473,600 2,187 Russia 1,540,360 931 Japan 1,224,740 1,110 India 1,007,980 651 Germany 837,425 607 UK 558,225 400 Australia 332,377 236 Canada 521,404 568 Brazil 327,858 386 Spain 304,882 278 Sketch a scatter diagram for the above data and interpret this graph. [4 marks] b) Assuming a linear relationship, use the least-squares method to find the regression coefficient, a and b. [8 marks] c) Interpret the meaning of the slope, b, in the problem. [3 marks] d) Predict the amount of CO2 emission for a country generating 1,000 terawatt hours of energy. [3 marks] e) Predict the amount of CO2 emission for a country generating 200 terawatt hours of energy. [2 marks] Page 5 of 5