MEASURES OF CENTRAL TENDECIES & DISPERSION

    Q1 (MEASURES OF CENTRAL TENDECIES & DISPERSION) The prices of a sample of 180 houses, in a northern area of a town, are as follows: Price band (£000s) Number of houses 140 to less than 160 30 160 to less than 170 21 170 to less than 180 60 180 to less than 200 20 200 to less than 250 25 250 to less than 350 24 (a) Estimate the mean and standard deviation of house prices in this area. [10 marks] (b) Give two brief reasons why your answer of the mean in (a) is only an estimate of the mean price of houses in the northern area. [5 marks] (c) Draw a cumulative frequency curve (ogive) for this data use the ogive to estimate: (i) The median value of the house prices [2 marks] (ii) The inter-quartile range of the house prices. [5 marks] (iii) The percentage of houses that are worth more than £215,000 [3 marks] (d) Calculate the following (i) The median value of the house prices [5 marks]   (ii) The inter-quartile range of the house prices. [5 marks] (iii) The percentage of houses that are worth more than £225,000 [5 marks] (iv) A prospective house buyer has a budget of only £155,000 but will be ready to up this to 5%. Estimate the percentage likelihood that the buyer can buy a house in this area. [5 marks] (v) If a buyer has a 65% likelihood, what budget should he/she need to have to purchase a house in the area. [5 marks] (e) Comment briefly on the difference between the median value and the mean calculated in (a). What does this tell you about the shape of the distribution of house prices? [5 marks] (f) A similar survey is conducted on the southern side of the town. The mean is estimated to be £205,000 with a standard deviation of £11,000. Write two brief sentences to describe the main differences between the northern and southern sides. [5 marks]   Q2 (MEASURES OF DISPERSION) The table below shows What Car December 2021 New Car Prices for nine models of cars. Alfa Romeo Audi BMW Citroën Mercedes Peugeot Porsche Renault Volvo 36.09 63.94 37.70 14.18 68.39 33.98 67.36 23.60 53.18 40.59 20.84 33.09 16.77 57.23 34.62 78.49 34.79 47.16 48.59 41.13 78.64 20.78 48.99 39.78 76.91 38.04 41.36 68.99 25.82 63.95 19.89 40.42 41.34 92.42 30.99 56.70 45.25 32.42 79.74 21.06 69.05 54.25 116.47 32.59 76.78 50.25 38.30 72.91 23.02 204.38 53.40 99.42 33.59 71.30 54.85 42.11 94.91 28.51 87.50 44.12 162.91 30.90 65.48 74.95 52.06 25.92 22.68 70.59 45.84 75.79 26.20 46.79 45.25 60.29 116.82 30.28 177.03 43.70 86.84 18.10 55.54 50.25 84.43 139.18 28.94 99.57 49.91 96.64 29.90 61.60 a) Calculate the following for each of the nine models (i) Using Excel, calculate the three averages and any three measures of dispersion. [10 marks] (ii) Using the coefficient of variation (CV), comment on the prices of the nine models. [10 marks] b) Using a class interval 7, width of 25 and the Excel frequency function, (i) construct a grouped frequency table of all the prices put together. [10 marks] (ii) Draw a histogram and comment on the shape of the diagram. [5 marks] c) Using the formula for calculating the Correlation Coefficient, calculate the degree of associations of the car prices between Audi and BMW and also Renault and Volvo. Comment on the results. [10 marks] Q3 (DECISION TREE) (a) Max Burg is considering setting up a restaurant. If he opens it in Castle Street, he believes there is an equal chance of making a profit of £150,000 or £100,000 in the first year, depending on whether the trade is high or low. If, however, he opens a site on the High   Street, then he believes he has a 70% chance of making a profit of £150,000 and a 30% chance of making a profit of £100,000 in the first year. Where should he open the restaurant if he wishes to maximise his expected profit over the first year, and what is the expected profit from his chosen course of action? [10 marks] (b) Max is concerned about the possibility of the UK being in recession by the time the restaurant would be opened. He thinks there is a 40% chance of recession. If there is no recession, then the situation is as described in part (a). However, if there is a recession, then he estimates that the Castle Street restaurant would result in £20,000 profit or £30,000 loss over the first year with equal probabilities. At the High Street site, if there is a recession, he estimates he would have a 40% chance of making a profit of £100,000, and a 60% chance of losing £50,000 in the first year. Draw a decision tree for this problem. Using your decision tree determine if Max should open a restaurant if he wants to maximise his expected profit over the first year, and if so, where and what is the expected profit from this chosen course of action? [25 marks] ( c ) Discuss the limitations of the decision analysis performed and whether the maximisation of expected profits is a sensible criterion for Max to use when making his decision? [10 marks]   Q4 (INDEX NUMBERS) Below is the RPI index data for U.K. from 2011 to date. JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC 2011 229.0 231.3 232.5 234.4 235.2 235.2 234.7 236.1 237.9 238.0 238.5 239.4 2012 238.0 239.9 240.8 242.5 242.4 241.8 242.1 243.0 244.2 245.6 245.6 246.8 2013 245.8 247.6 248.7 249.5 250.0 249.7 249.7 251.0 251.9 251.9 252.1 253.4 2014 252.6 254.2 254.8 255.7 255.9 256.3 256.0 257.0 257.6 257.7 257.1 257.5 2015 255.4 256.7 257.1 258.0 258.5 258.9 258.6 259.8 259.6 259.5 259.8 260.6 2016 258.8 260.0 261.1 261.4 262.1 263.1 263.4 264.4 264.9 264.8 265.5 267.1 2017 265.5 268.4 269.3 270.6 271.7 272.3 272.9 274.7 275.1 275.3 275.8 278.1 2018 276.0 278.1 278.3 279.7 280.7 281.5 281.7 284.2 284.1 284.5 284.6 285.6 2019 283.0 285.0 285.1 288.2 289.2 289.6 289.5 291.7 291.0 290.4 291.0 291.9 2020 290.6 292.0 292.6 292.6 292.2 292.7 294.2 293.3 294.3 294.3 293.5 295.4 2021 294.6 296.0 296.9 301.1 301.9 304.0 305.5 307.4 308.6 Your annual salary in 2011 is £23,500. To take care of inflations, your employer been increasing your salary by 2% p.a. up to 2020. NOTE: Please comment on each answers and solution that you provide (a) Determine the yearly RPI for each year by taking the average (mean) of the monthly data. Using this yearly RPI data, index your salary to 2012, and determine for each year if you are earning more or less in real earning. [15 marks] (b) By what percentage increase should your salary be in 2020 to bring your salary in line with inflation? What should be your salary in 2020? [10 marks] (c) Using a simple linear regression and the first three quarter RPI, estimate the monthly RPI for the last quarter of 2021. Hint: use the months Jan – Dec (1 – 12) as the independent variable and the RPIs as the dependent variable. [10 marks] (d) What percentage increase should your employer offer in 2021 to bring your salary in line with inflation and what is this salary? Total Crime (Y) Police Funding (X1) 25yrs old in High School (X2) Not High School (X3) College Education (X4) 1740 22 45 22 15 732 22 69 13 14 1504 30 49 15 13 1504 21 49 15 13 989 22 45 9 13 940 66 44 26 16 940 33 30 26 16 821 30 52 13 12 821 30 57 13 12 783 55 57 23 11 783 55 72 23 11 1324 33 44 22 16 1324 82 67 22 16 912 27 44 21 9 805 46 57 14 12 867 39 52 17 10 657 33 72 13 22 792 28 57 12 11 491 32 67 11 16 491 32 67 11 16 The variables (Y, X1 to X4) are as follows Y = total overall reported crime rate per 1 million residents X1 = annual police funding in $/resident X2 = % of people 25 years+ with 4 yrs. of high school X3 = % of 16 to 19 year-olds not in high school and not high school graduates. X4 = % of 18 to 24 year-olds in college Take Y as the dependent variable, and X1 – X4 as independent variables. Using excel, undertake the following a) Undertake Correlation analysis between the dependent variable and each independent variable. Comment on the results. [10 marks] b) Undertake Correlation analysis between pairs of the independent variable. Comment on the results. [10 marks] c.) Which of the independent variables most explained the variation in Total Crime? Which variables least explained Total Crime? [10 marks] d) Undertake a simple linear regression to write out the equation for each of the four independent variables. Comment on the equations. [15 marks] e) The city authorities want to reduce crime, what should they best do? [5 marks] f) In the year when the city authority increased police funding to $100 per resident, what will approx. be the Total Crime rate per 1 million residents? [5 marks]