ST130: Basic Statistics Faculty of Science, Technology & Environment School of Computing, Information & Mathematical Sciences Final Examination Semester I, 2018 Mode: Face to Face and Online Duration of Exam: 3 hours + 10 minutes Reading Time: 10 minutes Writing Time: 3 hours Total marks: 100 INSTRUCTIONS: 1. This exam has FIVE (5) questions of 20 marks each and all are compulsory. 2. Write your answers in the answer booklet provided. 3. Start each question on a new page. 4. Show all necessary working. Partial marks will be awarded for partially correct answers. 5. There are SEVEN (7) pages in this exam paper (including this cover page). 6. This exam is worth 50% of the overall mark. The minimum exam mark is 40/100. 7. You may use a NON PROGRAMABLE calculator. 8. Eton Statistical Table and Formula Sheet are provided. ST130 FINAL EXAMINATION Question 1 SEMESTER I, 2018 Start on a new page [4 + 6 + 4 + 6 = 20 marks] (A) A manager of a bank conducted a survey to gauge the views of all his 12000 customers on internet banking. Internet banking as explained to the customers would incur less banking fees. In this survey, a customer was asked whether they are interested in internet banking or not. The analysis states that 21% of the 300 customers interviewed said that they are interested. (i) What is the population of interest in this survey? (ii) What is the sample size used in this survey? (iii) Is the value 21% a parameter or a statistic? Explain your choice. (1 + 1 + 2 = 4 marks) (B) In a class of 20 students, the test result of each student (out of 10 marks) is listed below. A teacher needs to select a sample of 5 marks from the list to estimate the mean mark of the entire class. Student Number 1 Test Mark 2 (i) 2 3 3 8 4 4 5 8 6 2 7 7 8 3 9 6 10 11 12 13 14 15 16 17 18 19 20 5 3 3 4 6 4 5 8 8 6 2 Use the systematic sampling method to select the 5 samples. Suppose our first randomly selected sample is the mark for student number 1, list the five samples by stating which student is selected with the corresponding mark. (ii) Evaluate the estimated class mean mark. (4 + 2 = 6 marks) (C) A grouped frequency distribution of weights (in kg) of 50 pieces of luggage is given below: Weight (kg) Class Boundaries No. of pieces 7-9 10 -12 13 - 15 16 - 18 19 - 21 Total 6.5 - 9.5 9.5 - 12.5 12.5 - 15.5 15.5 - 18.5 17.5 - 21.5 2 8 14 19 7 50 Use the information above to answer the following: (i) What is the class width for the distribution? (ii) What is the upper class limit of the second class interval? (iii) Calculate the class mark or midpoint of the second class interval. (1 + 1 + 2 = 4 marks) Page 2 of 7 ST130 FINAL EXAMINATION SEMESTER I, 2018 (D) A local internet service provider takes a sample of 30 subscribers to find out the number of hours they spend ‘on-line’ in a particular month. Here are the results arranged in ascending order: 6 9 14 15 16 17 18 21 23 24 24 27 28 30 30 33 33 33 34 35 37 38 43 46 47 49 49 57 59 63 (i) Draw a stem and leaf plot for this data. (ii) Explain why it is likely that the underlying population of all users may have a normal distribution. (4 + 2 = 6 marks) Question 2 Start on a new page [5 + 4 + 7 + 4 = 20 marks] (A) Probability can be classified into three basic approaches or interpretations. (i) List the three approaches. (ii) In an experiment of tossing a coin 10 times, only 2 heads appeared, hence the probability of getting a head is 0.2. Which approach is used here? Explain briefly. (3 + 2 = 5 marks) (B) Classify the events below as simple or compound. Explain your choice. (i) Getting a head in tossing a coin. (ii) Getting an even number when rolling a die. (2 + 2 = 4 marks) (C) A marble is drawn from a bag containing 3 white, 2 red and 5 blue marbles. (i) Let A be the event drawing a red marble and B be drawing a blue marble. Are the events A and B mutually exclusive? Explain your answer. (ii) What is the probability that the marble drawn is blue or red? (3 + 4 = 7 marks) (D) Joe is playing a game of chance at the Hibiscus festival, costing $1 for each game. In the game two fair dice are rolled and the sum of the numbers that turn up is found. If the sum is seven, then Joe wins $5. Otherwise Joe loses his money. Joe plays the game 15 times. Find his expected gain or loss. Page 3 of 7 ST130 FINAL EXAMINATION Question 3 SEMESTER I, 2018 Start on a new page [8 + 4 + 4 + 4 = 20 marks] (A) The normal distribution is a very important and most widely used distribution in statistics. (i) List 3 properties of a normal distribution. (ii) Calculators last on average 4 years (48 months) before developing a non-accidental defect. This calculator life time is normally distributed with a standard deviation of 10 months. If a retailer is only willing to replace 2% of the calculators under guarantee, what length of guarantee (to the nearest month) should be offered? (3 + 5 = 8 marks) (B) Explain the terms confidence level and confidence interval. (C) A recent survey of 8 social networking sites has a mean of 13.1 and a standard deviation of 4.1 million visitors for a specific month. Find the 95% confidence interval of the true mean. Assume that the variable is normally distributed. (D) If the variance of a national accounting exam is 900, how large a sample is needed to estimate the true mean score within 5 points and with 99% confidence? Question 4 Start on a new page [4 + 8 + 8 = 20 marks] (A) Define null hypothesis and alternate hypothesis, and give an example of each. (B) An attorney claims that more than 25% of all lawyers advertise. A sample of 200 lawyers in a certain city showed that 63 had used some form of advertising. At 0.05, is there enough evidence to support the attorney’s claim? Use the P-value method. The question should be answered as follows: (i) State the hypothesis and identify the claim. (ii) Compute the test value. (iii) Find the P-value. (iv) Make the decision and summarize the results. (2 + 2 + 2 + 2 = 8 marks) (C) The average yearly earnings of male college graduates (with at least a bachelor’s degree) is $58,500. The average yearly earnings of female college graduates with the same qualifications is $49,339. Page 4 of 7 ST130 FINAL EXAMINATION SEMESTER I, 2018 Based on the results, can it be concluded that there is difference in mean earnings between male and female college graduates? Use the 0.01 level of significance. Male Female Sample mean $59,235 $52,487 Population standard deviation $8,945 $10,125 40 35 Sample size The question should be answered as follows: (i) State the hypothesis and identify the claim. (ii) Find the critical value(s). (iii) Compute the test value. (iv) Make the decision and summarize the results. (2 + 2 + 2 + 2 = 8 marks) Question 5 Start on a new page [8 + 6 + 6 = 20 marks] (A) The data shown below is for the car rental companies in Fiji for a recent year. Company A B C D E F Cars (in thousands), x 63 29 20. 19. 13. 8.5 Revenue (in millions), 7.0 3.9 2.1 8 2.8 1 1.4 4 1.5 y Using the 5% level of significance and r 0.982, test the significance of the correlation coefficient as follows: (i) State the hypothesis. (ii) Find the critical value(s). (iii) Compute the test value. (iv) Make the decision and summarize the results. (2 + 2 + 2 + 2 = 8 marks) (B) In an experiment on growth of plants and soil type the following results were obtained: Page 5 of 7 ST130 FINAL EXAMINATION SEMESTER I, 2018 Growth Soil Type Total Clay Sand Loam Poor 16 8 14 38 Average 31 16 21 68 Good 18 36 25 79 Total 65 60 60 185 Compute the test value to test the hypothesis that there is an association between growth of plant and soil type. (C) A marketing specialist wishes to see whether there is a difference in the average time a customer has to wait in a checkout line in three large self-service department stores. The times (in minutes) are shown in the table below: Store A Store B Store C 3 5 1 2 8 3 5 9 4 6 6 2 3 2 7 1 5 3 Using the ANOVA table below and at 0.05, carry out a test whether there a significant difference in the mean waiting times of customers for each store. Source Sum of squares d.f Mean squares F Between 25 2 12.5 2.7 Within (error) 69.5505 15 4.6367 Total 94.5505 17 END OF EXAM FORMULAE Page 6 of 7 ST130 FINAL EXAMINATION 1. k SEMESTER I, 2018 total population . number of samples 2. X X . n 3. P A B P ( A) P( B) P( A B). 4. E ( X ) X P( X ). 5. z X . s s X t 2 , d . f n 1. n n 6. X t 2 z 7. n 2 . E X 8. t , d . f n 1. s/ n pˆ p 9. z . pq / n 2 10. z X 1 X 2 1 2 12 n1 11. t r 22 . n2 n2 , d . f n 2. 1 r2 row total (column total) , (O E ) 2 12. 13.747, where E E grant total and d . f .N k 1, d . f .D N k . 2 Page 7 of 7