Statistics

Statistics


Statistics and Analytics for Decision Making Q.1 (30 marks) A researcher is examining the relationship between cinema occupancy levels in a large city and a number of possible explanatory variables. She gathers monthly data for 48 consecutive months on: · Cinema Occupancy Levels · Employment in the catchment area of the city · Consumer Sentiment · Number of second level school students on trips to the city · Weather conditions in the city area (measured by average daily rainfall) She defines 3 models as follows: Model 1 Y = α + β.X1 + β4.X4 where Y = Cinema Occupancy Levels and X1 = Employment, X4 = Consumer Sentiment Model 2 Y = α + β1.X1 + β2.X2 where Y =Cinema Occupancy Levels, X1 = Employment and X2 = Second level school students on trips to the city Model 3 Y = α + β1.X1 + β2.X2 + β3.X3 where Y =Cinema Occupancy Levels, X1 = Employment and X2 = Retail spending by second level school students on trips to the city and X3 = Weather conditions in the city The following is the result of the Regression and Correlation analysis: Table 1 Model R R squared R squared change 1 0.684 0.468 0.468 2 0.846 0.716 0.248 3 0.960 0.922 0.206 Table 2 Model Coefficient Significance Zero-order Correlation 1 Constant 43 0.041 Employment 0.352 0.008 0.628 Consumer Sentiment 0.148 0.125 0.605 2 Constant 31 0.048 Employment 0.314 0.024 0.628 Students’ Spending -4.12 0.022 - 0.643 3 Constant 29 0.221 Employment 0.288 0.031 0.628 Students’ spending -3.262 0.037 - 0.643 Weather 0.067 0.327 0.158 (a) Write a short note explaining the change in the predictive power of the model (using R2 in Table 1) as additional variables are added or removed in moving from Model 1 to Model 2 to Model 3(8 marks) (b) Explain the relevance of and interpret the meaning of the significance values shown in the 4th column in Table 2. Are they satisfactory?(9 marks) (c) Explain the meaning of the negative sign on the regression coefficient of Student’s Spending in Models 2 and 3, the positive sign on the regression coefficient of Employment in Models 1, 2 and 3 and the positive sign on the regression coefficient of Weather in Model 3. (5 marks) (d) Consumer Sentiment is dropped from the Model after Model 1. Discuss why this might be done.(8 marks) Q.2 (30 marks) (a) Explain why indices are a favoured way of presenting and analysing data rather than examining actual data. You should present data on a real variable (with at least 36 observations) to support your answer.(8 marks) (b) Retail Sales indices are published monthly in both Volume and Value format. Explain the difference in the information content of the two. You should provide data (with at least 36 observations) on Retail Sales in both Volume and Value format to support your answer.(8 marks) (c) Retail Sales (both Volume and Value) indices are published monthly in both Seasonally-adjusted and Unadjusted format. Explain the difference in the information content of the two. You should provide data (with at least 36 observations) on Retail Sales in both Seasonally-adjusted and Unadjusted format to support your answer.(8 marks) (d) Explain why moving average data is sometimes used in preference to actual data in order to find the underlying ‘message’ or trend in a data set. You should provide data showing actual data and moving-average data (of the same variable) to support your answer.(6 marks) Note: Different data sets and variables should be used for each part above, apart from Retail Sales in b) and c). Part B (30 marks) Please answer 2 of the 3 questions in this section Q.3 Answer 3 of the following 5 parts (a) – (e).(15 marks) (a) Explain the difference between Stratified random sampling and Cluster sampling. (5 marks) (b) Explain what is meant by multicollinearity in regression analysis(5 marks) (c) Explain the difference between a leading and a lagging variable in regression analysis. (5 marks) (d) Explain the difference between positive and negative correlation and give an example of each.(5 marks) (e) Explain why it might be useful to examine a non-linear format rather than (or in addition to) a linear format in regression and correlation analysis.Give an example of a relationship that might be fairly strong but non-linear. (5 marks) Q.4(15 marks) The value of houses in a certain part of Dublin is normally distributed with a mean of €480,000 and a standard deviation of €140,000. (a) What proportion of houses have a value greater than €550,000(3 marks) (b) What proportion of houses have a value less than €760,000(3 marks) (c) What proportion of houses have a value in the range €410,000 - €690,000? (4 marks) (d) What value cuts off the 16% least valuable houses(5 marks) Q.5 (15 marks) Note in answering question 5, you must provide a clear explanation of what the result means. (a) A random sample of 81 pubs in Dublin city centre showed that the average cost of a pint of Triple Black beer was €5.80. The standard deviation of the sample was €1.60. Construct a 95% confidence interval for the average cost of a pint of Triple Black beer in all Dublin city centre pubs.(6 marks) (b) (i) A survey of 500 recent business graduates of full-time college courses in Ireland showed that 360 got jobs within 4 months of graduating. Construct a 99% confidence interval for the proportion of all recent business graduates of full-time college courses in Ireland that got jobs within 4 months of graduating. (6 marks) (ii) Based on the confidence interval, is it likely that at least two-thirds of all recent business graduates of full-time college courses in Ireland got jobs within 4 months of graduating. (3 marks) Part C (40 marks) Both questions in this section are compulsory. You have the choice to answer (a) or (b) in each of these questions Note in answering questions 6 and 7, you must provide a clear explanation of what the result means. Q.6 Please answer part (a) or part (b) below. (20 marks) (a) A restaurant owner is selling her business and claims that the average amount spent on a meal per person in the restaurant is €100. A potential buyer requests that he be permitted to examine recent bills to check out this claim. A random survey of 140 bills shows an average spend per person on a meal of €97 with a standard deviation of €16. Using a 1% test of significance, does this imply that the average spend per person on a meal is less than €100?(20 marks) (b) A trade union official who is a member of a company’s Health and Safety Committee claims that 48% of all workers in the company have received injuries at work over the past two years which has caused them to take time off work. The Chairperson of the Committee decides that this claim should be checked. A random sample of 600 workers showed 270 had received injuries at work over the past two years which has caused them to take time off work. Using a 5% significance test, does this evidence suggest that the proportion of all workers who received injuries at work over the past two years which caused them to take time off work is lower than 48%?(20 marks) Q.7 Please answer part (a) or part (b) below.(20 marks) (a) Higher College Ireland has a very large first year Marketing class, with a total of 300 students. Two different lecturers, Bill and Gill, each mark 150 of the first year exams. The Dean of the School of Business wishes to check if the two lectures marks are approximately the same, i.e., they give roughly equal marks for similar quality examination answers. To test this, the Dean asks the two lecturers to mark the same sample of 12 exam scripts. The table below shows the marks awarded by Bill and Gill for each of the 12 scripts The standard deviation of the difference in the marks awarded by Bill and Gill is 3.315 Using a paired (dependent) sample test, test the hypothesis, (at a significance level of 5%), that the average difference in the marks awarded by Bill and Gill is not zero, i.e. there is a difference in the average mark awarded. The population standard deviation is unknown.(20 marks) (b) The Head of Costumer Relations for an international Department Store chain is examining customer satisfaction at two different stores. She commissions an independent survey to examine the issue. A sample of 400 customers of the Southside store showed that 320 were satisfied with the service in the store while a sample of 300 customers in the Northside store showed that 225 were satisfied with the service in the store. The two samples are independent. Perform a hypothesis test at the 1% level of significance, to determine whether the proportion of customers of the Southside store who are satisfied with the service in the store is higher than the proportion of customers in the Northside store who are satisfied with the service in the store. (20 marks) Statistics and Analytics for Decision Making BA Business Management Statistics and Analytics for Decision Making Notation, Formulae and Tables Notation Population Sample Average/Mean μ _ X Standard Deviation ơ s Proportion π p Size N n Number of ‘Successes’ X Formulae Estimating and Confidence Intervals: 1. Confidence interval for a population mean (Population standard deviation ơ known) _ μ =X +/- Z x S.E. σ Standard error of the mean (S.E.)=--- √n σ =population standard deviation n= sample size 2. Confidence interval for a population mean (Population standard deviation ơ unknown and sample size > 30) __ μ = X +/- Z x S.E. s Standard error of the mean (S.E.)=--- √n s =sample standard deviation n= sample size 3. Confidence interval for a population proportion π =p +/- Z x S.E. Standard error of the sampling distribution of a sample proportion (S.E.) = __________ / p x (1 – p) \/ n p =proportion of ‘successes’ in sample = X/n X= number of ‘successes’ in sample n= sample size One-sample tests of Hypothesis: 4. Test statistic for hypothesis test of