BUS105 Statistics. Tutor Marked Assignment. Total Marks: 45; Weightage: 15%

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1 BUS105 Statistics Tutor Marked Assignment Total Marks: 45; Weightage: 15% Objectives a) Reinforcing your learning, at home and in class b) Identifying the topics that you have problems with so that your lecturer can help you c) Building up your confidence for the final examination d) Applying statistical competency in real life scenarios Marks allocation Most questions have more than 1 part. 1 mark for each part of the question. Study Unit 1 Study Unit 2 Study Unit 3 Study Unit 4 Study Unit 5 Study Unit 6 Total Question Marks Awarded Page 1 of 12

2 Instructions a) Arial, Font Size 12, 1.5 spacing. b) Word limit: 1,000 words c) You may answer in point form. d) Workings not required. e) Incorporate output from statistical runs where appropriate. f) Title page (available from Canvas course site) g) Start a new question on a new page, where appropriate. Submission Submission of the TMA has 2 components: a) Submit the assignment electronically by am Mar 7. b) Drop hard-copy of assignment into one of the 2 black boxes on either side of C9.03 by 6 pm on Mar 7. Page 2 of 12

3 Study Unit 1 Question 1 (3 marks) A scatter diagram is commonly employed to describe the relationship between two interval variables: a dependent variable Y; and an independent variable X. a) Explain the concepts of dependent variables and independent variables with an example, other than the textbook examples and examples the lecturer explained in class. b) When Y is related to X, can we say X is causing Y? Please explain with an example. c) What are the two aspects that a scatter diagram can tell of the relationship between two interval variables? Question 2 (1 mark) After your BUS105 examination, you are advised that your score is at the 70 percentile. You told your parents that you had score 70% in your BUS105 examination. Please explain with an illustration whether your advice of your BUS105 performance to your parents is accurate. Page 3 of 12

4 Question 1 Study Unit 2 (3 marks) Company ABC has submitted bids on two construction contracts, successively. The company believes that there is a 40% probability of winning the first contract. If the company wins the first contract, the probability of winning the second contract is 70%. If the company loses the first contract, the probability of winning the second contract is 50%. Hint: Use the decision tree technique to answer the question. a) What is the probability that Company ABC would win both contracts? b) What is the probability that Company ABC would lose both contracts? c) What is the probability that Company ABC would win either contract? Question 2 (1 mark) You are the owner of the car park at Mall ABC, and you are interested to know how long your customers have parked their cars in your car park. On a particular day, you select 250 cars at random, and record their duration of parking as follows: Number of Hours Number of Cars Page 4 of 12

5 You select a car at random. What is the probability the duration of parking is less than 3 hours? Hint: Convert the data on number of hours parked to a probability distribution first. Study Unit 3 Question 1 (2 marks) The weightage of the final examination in BUS105 is 50% The population mean and standard deviation of final examination marks were found to be 30% and 10%, respectively. a) What is the probability that a student s mark in BUS105 would be less than 22%? b) The probability of getting A+ is 1%. What is the minimum mark in the final examination that a student has to achieve to warrant an A+? In another word, what is the mark such that the probability of getting it is only 1%? Question 2 (2 marks) A statistic professor is investigating how many classes the BUS105 students miss each semester. The professor took a number of random samples of 100 BUS105 students, and checked against the attendance records to identify how many classes the students had missed in the previous semester. The sample mean of sampling distribution of the number of classes missed is The population standard deviation of number of classes missed is estimated to be 2.2 classes. a) Determine with 99% confidence level the confidence interval estimator of the number of classes BUS105 students miss. b) How would you interpret the confidence interval estimator? Page 5 of 12

6 Study Unit 4 Question 1 (8 marks) Café ABC claims that the daily sales of coffee is 100 cups. The taxation office suspects that Café ABC reports a lower than the actual sales figure in its annual tax return. As part of the field investigation, a taxation officer went to the café for 150 days at random, and counted the cups of coffees sold on each of these days, respectively. The mean from the sampling distribution was found to be 104. From other equivalent cafes, the taxation officer estimated that the population standard deviation of daily coffees sold was 22. a) Why would the taxation officer use z-statistic in the statistical analysis? b) Please set up the H0 and H1 hypothesis. i) H0; ii) H1. c) Is the test left-tail, right-tail, or two-tail? Please explain. d) What is the critical value of z at 5% significance? e) What is the z value from the sample? f) At 5% significance, would the taxation officer conclude that there is enough evidence that the daily sales of coffee is greater than 100? Please explain. g) What is the p-value from the analysis? h) What is the taxation officer s conclusion from the p-value? Please explain. Page 6 of 12

7 Question 2 (7 marks) You own a bakery. On average, 100 loaves of bread have been baked daily. Recently, you have hired a new baker, and you suspect that the output has dropped. So you decide to investigate the matter with statistical analysis. You noted the loaves of bread baked daily from 15 days at random, respectively, and calculated the following: x = S 2 = a) Why do you have to use t-statistic in ascertaining whether the output has decreased? b) Please set up the H0 and H1 hypothesis. i) H0; ii) H1. c) Is the test left-tail, right-tail, or two-tail? Please explain. d) What is the critical value of t at 5% significance? e) Calculate the value of the t-statistic. f) At 5% significance, would you conclude there is enough evidence that the mean daily output has decreased since the employment of the new baker? g) What do you think the p-value of the t-test would be: greater than or less than 0.05? Please explain. Page 7 of 12

8 Question 1 Study Unit 5 (4 marks) Do men listen more music than women? Or the other way round? Based on a survey, it is found that, the standard deviation of the number of minutes men and women listening to music daily are 10 minutes and 12 minutes, respectively. Assume the critical value is Conduct a hypothesis test to ascertain whether there is a difference in the variations of music listening times between men and women. a) State H 0 and H 1. b) Specify the distribution of test statistic. c) Compute the test statistic. d) What is your conclusion? Please explain. Page 8 of 12

9 Question 2 (4 marks) One group of students (Group A) learns statistics with Method A; another group (Group B) learns with Method B. Both group of students take the same examination afterwards. The examination marks of a sample of 6 students from Group A are: 19, 17, 23, 22, 17 and 18. The examination marks of a sample of 8 students from Group B are: 32, 28, 31, 26, 23, 24, 27 and 25. Can we infer that both Method A and Method B are just as effective in teaching statistics, or one method is more effective than the other? Assume everything else is equal, students diligence and competence, etc. Conduct a hypothesis testing to confirm whether the two mean scores are equal, at 5% significance. a) State H 0 and H 1. b) Ascertain the critical value, and specify whether it is a one-tailed or two-tailed test. c) Use Excel or other software to run ANOVA to ascertain the test statistic. d) What is your conclusion? Please explain. Page 9 of 12

10 Study Unit 6 Question 1 You are the CEO of a company that markets solar power panel. disappointing. (10 marks) Sales has been Marketing Director has been complaining that not enough money has been spent on advertising. You are not sure whether advertising is an effective strategy to enhance sales. Accordingly, you conduct a statistical investigation. You divide the country into 26 districts. You then collect data on the following variables: Sales ( 000s); advertising dollars ( 000s); number of active accounts; number of competitors; and market potential of each district. You aim to analyse and predict sales with a multiple regression model. a) Based on the following scatter plots, comment on the relationships between sales (dependent variable) and respective independent variables (advertising, accounts, competitors, and market potential). Scatterplot of Sales vs Advertising, Accounts, Competitors, Potential Advertising Accounts Sales Competitors Potential Page 10 of 12

11 b) Ascertain the degree of multicollinearity (degree of correlation between independent variables among one another) from the following correlation matrix. Pearson correlations Sales Advertising Accounts Competitors Advertising Accounts Competitors Potential c) Develop the regression equation from the given multiple regression output: Predictor Coef SE Coef T P Constant Advertising Accounts Competitors Potential S = R-Sq = 98.9% R-Sq(adj) = 98.7% Analysis of Variance Source DF SS MS F P Regression Residual Error Total d) Conduct a global test on whether all the regression coefficients of the independent variables are zero. Assume 5% significance. List all the relevant steps. e) Interpret the regression coefficients of the independent variables. Note: Sales and advertising dollars are in thousands (000 s). f) Conduct a hypothesis test on whether the individual regression coefficients of the independent variables are 0, with t statistic as well as p value. Assume 5% significance as in d). g) The multiple regression model has been improved with stepwise regression. Develop the respective regression equation from the following regression output. Regression output and equation without market potential as a variable: Page 11 of 12

12 Predictor Coef SE Coef T P Constant Advertising Accounts Competitors Regression output and equation without advertising as a variable: Predictor Coef SE Coef T P Constant Accounts Competitors h) Why do we need adjusted coefficient of determination ( R adj ) when we already 2 have coefficient of determination ( R )? Please explain. i) Please comment on whether the multiple regression model [from c)] is a good model based on the histogram of residuals. 9 Histogram of the Residuals (response is Sales) Frequency Residual j) For the multiple regression model with the regression equation Sales = Accounts Competitors in c), the variance inflation factor (VIF) is 1.1. Please comment on whether the multiple regression model [from c)] is a good model. Page 12 of 12