MAT10251 Statistical Analysis Project Of Used Car Assessment Answer

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Question :

MAT10251 STATISTICAL ANALYSIS PROJECT 

This project leads you through a statistical analysis of used car data. The data for this project was obtained from the car sales website  www.carsales.com.au  2 to 10 January 2019 (inclusive).

Part A covers parts of Topics 1 and 2, Part B parts of Topics 5 to 9.

You will need to work on this project throughout Session 1.

Project Data

The data for this project can be accessed from the MySCU site for MAT10251 in Task 2 - Project under ASSESSMENT

The data set provided contains 10 randomly chosen samples of size 115.

To obtain your data

(1)Click on the Project Data file. This will download an Excel file.

(2)Select the 7 columns (Year to Price) of data for the sample specified by the last digit of your student ID number. 

(3)Copy this into a new Excel file.

There are 10 sample data sets each of 7 columns (Year to Price

Your sample number matches the last digit of your SCU student ID number. For example, if your student ID number ends in 2 your sample is Sample 2 and you will be analysing used car data for Toyota Corolla cars for sale in Western Australia in columns Q2:W120.


Project Situation

An online consumer group Oz-Price-Watch regularly analyses used car prices in various Australian states. 

As a research assistant for Oz-Price-Watch, you are analysing the data for the Car and state specified by your sample. For example, if your student ID number ends in 0 your sample is Sample 0 and you will be analysing prices of used Toyota RAV4 (4 cylinder) in New South Wales.

You are required to analyse your sample data in response to the given questions and provide a written answer.  You can assume that your written answers are components of a longer report on used car prices.


Project Preparation

You are expected to use Excel when completing the project.

Your written answers presenting your findings and conclusions should be considered as a part of a larger report on used car prices. Each written answer should be a word document into which your Excel output has been copied

In addition, your statistical workings for Part B should appear as appendices to your written answer. This should include all necessary steps and appropriate Excel output.

Each part of the project should be submitted as a SINGLE Word document, with appropriate Excel output added.

Notes

  • You should not need to read beyond the study guide and textbook to complete the project.


Referencing

You are not required to reference. 

However, as the format of your written answers are components of a longer report it may be appropriate to reference. In this case, use any consistent referencing style.

Furthermore, you are not required to use real references. That is, any reference can be fictitious/fake. 

You are not required to reference any output or text from Part A that you reuse in Part B.


Project Submission

  • Each part of the project should be a SINGLE Word file with Excel output included. 
  • The given cover sheets should be the first pages of your submitted project and are not part of the page limit. 
  • DO NOT submit your appendices, which are not part of the page or word limit, for Part B as a separate file. 
  • Ensure that the page setup of your submitted document is A4 Portrait, with an appropriate format so that it is easily readable if printed.
  • Use line spacing of at least 1.5.
  • Please name your file 

“Family Name_First Name_Part_A/B/_Campus”

  • For example; Jayne_Nicola_Part_A_Lismore


Penalties For

Incorrect Sample

  • If you use a sample that does not correspond to the last digit of your student ID number, to be entered on the cover sheet, a maximum of two marks may be deducted, as this causes the marker extra work and frustration.


Incorrect Format 

  • If the page setup of your submitted Word file is not as required (that is, A4 Portrait, with appropriate format so that it is easily readable if printed), with at least 1.5 line spacing or your project is not submitted as a single Word document a maximum of two marks may be deducted, as this causes the marker extra work and frustration.
  • If your submitted file is not a Word file, for example it is a pdf or a zip file, a maximum of two marks may be deducted, as this causes the marker extra work and frustration.
  • In addition, if your file is not named as requested or the required cover sheets are not included or correctly completed a maximum of two marks may also be deducted, as this can cause the marker extra work and frustration.


MAT10251 STATISTICAL ANALYSIS

PROJECT - PART A


Due Week 4 Tuesday 26 March 2019

If you are a late enrolment in MAT10251, email Nicola Jayne nicola.jayne@scu.edu.au with the date you enrolled in MAT10251 for a revised due date

Value: 10% 

Objectives:1 to 5

Topics:1 and 2

Purpose:To 

  • introduce you to the project data, situation and Excel
  • use Excel to graph data and calculate statistics
  • interpret and communicate Excel results


Part A Preliminary Analysis of Sample Data

Oz-Price-Watch has asked you for a preliminary analysis of your sample data. Your calculations and conclusions from this analysis may be incorporated in your answer for Part B


Tasks – Part A Submission

Complete the following

1)Download and save your data.

2)Download the Project Part A cover sheets, name and save this file as

 “Family Name_First Name_Part_A_Campus”

3)Enter your Sample Number on page 2 of the Part A coversheets.

4)Statistical Answers: For used cars of the make and model for sale in the state specified by your sample perform the following 

Price of two and three year old cars

Using Price (7th column of data) explore prices of 2016 and 2017 used cars, by using Excel to:

  • Construct a frequency histogram or polygon for the price of two and three year old cars.
  • Calculate descriptive statistics for the price of two and three year old cars.

Note: The required data for 2016 and 2017 used cars is in the first rows of your sample.

Difference in price between cars for sale privately and those for sale by a used car dealer.

Use Price (7th column of data) and Seller (5th column of data), where Private indicates a private sale and Dealer a sale through a used car dealer, for all 115 cars in your sample to explore if there is a difference in price between the samples by using Excel to:

  • Construct separate boxplots, on the same plot or separately, for private sale prices and for used car dealer prices. 
  • Calculate descriptive statistics for private sale prices and for used car dealer prices. 

Hint: Sort data on Seller to obtain two samples. That is, price of used cars sold privately and price of used cars sold through a used car dealer.

Relationship between price and age and between price and odometer reading 

Explore the relationship between the price of a used car and its age and also the price of a used car and its odometer reading, by using Age (2nd column of data) and Odometer (3rd column of data) as independent variables with Price (7th column of data) as the dependent variable for all 115 cars in your sample, by using Excel to:

  • Construct scatter plots for Age and Price and for Odometer and Price
  • Calculate the correlation coefficient for Age and Price and for Odometer and Price.

5)Written Answer – Preliminary Analysis

Using the instructions given on pages 4 and 5 of the Part A coversheets, introduce your data and the results of your preliminary investigation of the price of used cars, of the make and model in the state specified by your sample.

This should be three to five pages and 400 to 800 words.

Use an appropriate style, without statistical jargon and equations, to clearly communicate your results. 

6)Complete Coversheets 1 and 2, save and submit Part A of the project online using Project Part A link in Submit Project by the due date Tuesday 26 March 2019.

Statistical Calculations

  • To obtain full marks your graphs and plots must be correct, including correct labels on both axes and a title. 

Marks will be deducted if:

  • Graph or plot incorrect

Examples

  • Gaps between classes of non-zero frequency in a histogram for continuous data
  • Incorrect independent and dependent variables in a scatter plot.
  • Line graph instead of a histogram 
  • Excel, PhStat, Histogram Workbook, or similar, is not used.
  • Axes incorrectly or not labelled.
  • No title.
  • For a histogram or frequency polygon inappropriate classes are used.
  • Scale on axes distorts graphs.
  • To obtain full marks for descriptive statistics copy the output table of the Descriptive Statistics command in Data Analysis or Descriptive Summary and/or Boxplot command in PhStat or Descriptive workbook. You may delete unnecessary statistics in these tables.
  • Marks will be deducted if any descriptive statistics are incorrect, so check:
  • Your sample size. 
  • Whether you are calculating sample statistics or population parameters.

Written Answer – Preliminary Analysis

  • 400 to 800 words and three to five pages - marks will be deducted if this is greatly exceeded. 
  • To obtain full marks must:
  • Be well structured.
  • Clearly communicate the results of the Excel output in language appropriate for your audience.
  • Include appropriate graphs and plots with appropriate statistics.
  • Provide information on average price of two and three-year-old used cars, how prices vary and any pattern to prices. 
  • Provide information on any difference in price of cars for sale privately and through a used car dealer
  • From your scatter plots discuss any apparent relationship between age and price, and also odometer reading and price. Comment on the strength, shape and sign of the relationships.
  • Marks will be deducted if:
  • There is little or no comment on, or interpretation of, the Excel output.
  • Unnecessary statistical jargon and equations appear.
  • It is confusing or not readable.
  • It is handwritten.
  • For each major spelling and/or grammatical error half a mark will be deducted, up to a maximum of two marks.
  • Also up to two marks may be deducted for poor structure and/or presentation.


MAT10251 STATISTICAL ANALYSIS

PROJECT – PART B


Due: Week 11 Sunday 19 May 2019

Value: 25% 

Objectives:1 to 5

Topics:5 to 9

Purpose:To apply your knowledge of statistical inference and regression to answer questions about used cars for sale by analysing the data and communicating the results.

Part B Submission

You should submit a single word document consisting of:

  • Part B coversheets
  • Written answer as components of a report. This should follow the format given on pages 4 and 5 of Part B coversheets
  • Appendices for Part B which contain full statistical working for the required statistical tasks.  This should follow the format given on pages 5 and 6 of Part B coversheets

Part B Preparation

The graphs, plots and interpretations in Part A may be required in the statistical and written answers in Part B.  Therefore, check these and make any required corrections.

While the submission date for Part B is Sunday 19 May 2019, you should be working on Part B during Weeks 6 to 11.

It is recommended that you follow the following timetable:

  • Question 1, covering Topic 5, should be completed in Week 6
  • Question 2, covering Topic 6, should be completed in Week 8
  • Question 3 covering Topic 7 should be attempted in Week 9
  • Question 4 covering Topic 8 should be attempted in Week 10
  • Question 5 covering Topic 9 should be attempted in Week 11

Task 1 Part B - Appendices Statistical Inference and Regression and Correlation Tasks (38 marks)

The following statistical tasks should appear as appendices to your written answers. These should include all necessary steps and appropriate Excel output. 

These appendices should come after your written answer within your single Word document for Part B.

Statistical Inference 

Choose a level of significance for any hypothesis tests and a level of confidence for any confidence intervals. Enter these values on page 2 of the Part B coversheets along with the sample number from Part A.

For used cars of the make and model for sale in the state specified by your sample answer the following questions using appropriate statistical inference and regression techniques.

Question 1 – Topic 5 (5.5 marks)

Since many buyers wish to purchase a two or three year old used car Oz-Price-Watch has asked you to provide information on the average price of 2016 and 2017 cars of the make and model for sale in the state specified by your sample.

To enable you to answer this use Price (7th column of your data) for 2016 and 2017 cars only, your output from Part A and an appropriate statistical inference technique to:

Estimate the population mean price of two and three year old used cars of the make and model for sale in the state specified by your sample

Note: The required data for 2016 and 2017 cars is in the first rows of your sample.

Question 2 – Topic 6 (7.5 marks)

Many buyers believe that white cars are safer since they are more visible. Therefore, they wish to purchase a white car. Oz-Price-Watch has asked you to explore if restricting a purchase to white cars will limit a buyer’s choice.  Past research by Oz-Price-Watch has shown that if a search is restricted to a feature, for example colour or transmission, which at most 30% of cars for sale have then buyer choice is limited.

To provide a justified answer to the question use White (6th column of data, where Yes = car for sale is white and No = car for sale is not white) for ALL 115 cars in your sample and an appropriate statistical inference technique to answer the following question

Are more than 30% of used cars of the make and model for sale in the state specified by your sample white?

Hint: Sort data on White to enable you to easily count the number of white cars in your sample.

Question 3 Topic 7 (8 marks)

Oz-Price-Watch wishes to know if there is a difference in price between cars for sale privately and those for sale by a used car dealer.

To provide a justified answer to this question use Price (7th column of data) and Seller (5th column of data) for all 115 cars in your sample, your output from Part A and an appropriate statistical inference technique to answer the following question 

Is there a difference in the average price of cars, of the specified make and model for sale in the specified state, for sale privately and by a used car dealer?

Hint: Sort data on Seller to easily obtain two samples – Prices for private sellers and for used car dealers.

Questions 4 and 5 Simple and Multiple Linear Regression (17 marks)

Oz-Price-Watch asks you how the value of a used car, of the specified make and model, depreciates.

To answer this you develop a simple linear regression model to predict price from age or odometer reading and a multiple linear regression model to predict price from age, odometer reading and transmission type. Then, to provide a justified answer to Oz-Price-Watch, choose and interpret the linear model that best fits your data. 

Question 4 Simple Linear Regression Model Topic 8

From your results in Part A choose either Age or Odometer as an independent variable, to predict Price.

To explore the relationship between the age or odometer reading of a used car and its price, use your output from Part A and Age or Odometer (2nd or 3rd column of data) as an independent variable with Price (7th column of data) as the dependent variable, for all 115 cars in your sample, to develop and then explore a simple linear relationship between the two variables by:

  • Calculating the least squares regression line, correlation coefficient and coefficient of determination.
  • Interpreting the gradient and vertical intercept of the simple linear regression equation.
  • Interpreting the correlation coefficient and coefficient of determination. Are these values consistent with your scatter plot?

Note: You can choose either Age or Odometer as the independent variable in this model.

Question 5 Multiple Linear Regression Model Topic 9 

To explore what other factors may have an influence on the value of a used car use your output from Part A and Age, Odometer and Transmission (2nd, 3rd and 4th columns of data) as three independent variables with Price (7th column of data) as the dependent variable for all 115 cars in your sample, to develop and then explore the relationship between these four variables by:

  • Calculating the multiple regression equation, multiple correlation coefficient, and coefficient of multiple determination.
  • Interpreting the values of the multiple regression coefficients.
  • Interpreting the values of the multiple correlation coefficient and coefficient of multiple determination. Compare these values with the corresponding values for the simple linear regression model.

Then determine the best model to predict the price of a used car by:

  • Using appropriate tests to determine which independent variables make a significant contribution to the regression model.
  • Give or calculate the simple or multiple regression equation which best fits the data. 

Notes:

  • You may need to transform or manipulate the given data, before using Excel for the corresponding statistical calculations.
  • Use Excel for all statistical calculations. You do not need to repeat any Excel calculations by hand. However, make sure that you define your random variables and include any steps not given by Excel. For example, in a hypothesis test include the null and alternative hypotheses, along with the decision to reject or not reject the null hypothesis.
  • Mention any assumptions you need to make, where appropriate justify these from Part A output.
  • In Question 4 fit a linear model even if from your scatter plot you decide that a non-linear relationship better fits the data or that no apparent relationship exists. However, mention this in your written answer and/or corresponding appendix.
  • Comment on why a test or confidence interval has been chosen. Where appropriate include and refer to Part A output.
  • Make sure you interpret confidence intervals and write conclusions to hypothesis tests.

Task 2 - Written Answer – Components of a report (12 marks)

For Questions 1, 2, 3 and Questions 4 and 5 combined present the results of your calculations, with your interpretation and conclusions as components of a longer report on used car prices.

Use the instructions given on pages 4 and 5 of the Part B coversheets.

This should be 500 to 1100 words and three to seven pages.

It should be submitted as a Word file with Excel output included. 

Make sure you:

  • Introduce each question and put it in context
  • Answer each question in non-statistical language.
  • Present the result of your calculations and tests without unnecessary statistical jargon
  • Include a conclusion which answers the given question. 

In particular, for Questions 4 and 5

  • Mention or explain your choice of independent and dependent variables
  • Include and justify the best model.
  • Discuss and interpret the values of the regression and correlation coefficients of the best model.
Show More

Answer :

Q1: Pivot table was used to generate average prices for 2016 and 2017. Further, apart from the overall average prices, averages were also calculated for seller type and transmission type as reflected in following table:

Row LabelsAverage of Price
2016              14,919.84 
Dealer              14,840.00 
Automatic              14,797.14 
Manual              14,990.00 
Private              14,991.70 
Automatic              15,109.50 
Manual              14,913.17 
2017              17,091.53 
Dealer              17,041.11 
Automatic              17,102.41 
Manual              15,999.00 
Private              17,999.00 
Automatic              17,999.00 
Grand Total              16,005.68 


It can be seen that overall average for 2016 is $14,919.84 and for 2017 is $17,091.53. 

Q2: From given data for Mitsubishi make cars in Queensland, we found out number of white and non-white cars:

ColourNumber%
Non White7464.3%
White4135.7%


Hence the research hypothesis can be said to be whether the proportion of white cars of Mitsubishi make in Queensland is more than 30%.

We will use z-test for a proportion.

Hence, from above, p^= 0.0357% where n = 115

Assuming α = 0.05, we can state null and alternative hypothesis as:

Null hypothesis H0: p > 0.30

Alternative hypothesis H1: p ≤ 0.30

We will calculate z-statistic and find corresponding value form z-table:

Ƶ = (0.357-0.30)/√(0.30*0.70)/115 = 1.3339

From z-table, the significance level is p = 0.1822 which is greater than our significance level of p = 0.05. Hence, we are unable to reject the null hypothesis. 

We can conclude that at significance level of p = 0.05, there is statistically significant evidence that white cars account for more than 30% of the total cars available of Mitsubishi make in Queensland.

Q3: From given data for Mitsubishi make cars in Queensland, we found out number, average price and standard deviation in prices of cars sold by dealer and private sellers:

SellerNumberAverage PriceSD
Dealer71       12,408.93          4,243.81 
Private44       11,325.45          4,055.51 


Hence the research hypothesis can be said to be whether the average price of cars sold by dealers is equal to price of cars sold by private sellers for Mitsubishi make cars in Queensland.

Assuming α = 0.05, we can state null and alternative hypothesis as:

Null hypothesis H0: µ1 = µ2

Alternative hypothesis H1: µ1 ≠ µ2

We will use t-test for Two-Sample assuming equal variances in excel. The output is as follows:

 t-test for Two-Sample

From above we can see that the p-value is 0.1787 which is greater than our significance level of p = 0.05. Hence, we are unable to reject the null hypothesis. 

We can conclude that at significance level of p = 0.05, there is no statistically significant evidence to reject null hypothesis. Hence, we can conclude that the average price of cars sold by dealers is equal to price of cars sold by private sellers for Mitsubishi make cars in Queensland.

We get same result if we use the test assuming unequal variances:

unequal variances

From above we can see that the p-value is 0.1746 which is greater than our significance level of p = 0.05. Hence, we are unable to reject the null hypothesis. 

We can conclude that at significance level of p = 0.05, there is no statistically significant evidence to reject null hypothesis. Hence, we can conclude that the average price of cars sold by dealers is equal to price of cars sold by private sellers for Mitsubishi make cars in Queensland.

Q4:

Simple linear regression

The dependent variable (y), Car price was regressed on the independent variable (x), Odometer reading by using Regression in MS-excel:

The dependent variable

It can be seen in above output that value of R is 0.83 and R2 is 0.69 indicating that model is a reasonably good fit with around 69% of the change in car price being attributed to the odometer reading. We can interpret the least squares regression equation as follows:

y= 17,388.03 – 0.06 x1 

where,

  • Dependent variable, Car price is represented as: y
  • Independent variable, Odometer reading (kms) is represented as x1 

We can see that there is inverse relationship between the variables such that as odometer reading increases, car price decreases and vice versa. The above equation has a constant of 17,388.03 which is also known as Y-intercept coefficient.

 The p-values indicate statistical significance of various variables. We can see that the p-value for Odometer reading is p = 0.0000 which is less than p = 0.05. Hence, this variable is statistically significant.

Further, from confidence interval data, we can be 95% confident that with each increase in odometer reading (kms), the car price will decrease between -0.07 and -0.05.

Q5:

Multiple linear regression

The dependent variable (y), Car price was regressed on the independent variables (x), Age, Odometer reading (kms) and Transmission, by using Regression in MS-excel:

Multiple linear regression

In above, we had to convert Transmission data to numeric data by using following connotation: Automatic was denoted as ‘1’ and Manual was denoted as ‘0’.

It can be seen in above output that value of R is 0.92 and R2 is 0.84 indicating that model is a very good fit with around 84% of the change in car price being attributed to the age, odometer reading (kms) and transmission. We can interpret the least squares regression equation as follows:

y= 17,163.31 – 625.68 x1 – 0.02 x2– 1,301.91 x3 

where,

  • Dependent variable, Car price is represented as: y
  • Independent variable, Age is represented as x1 
  • Independent variable, Odometer reading (kms) is represented as x2 
  • Independent variable, Transmission is represented as x3 

The above equation has a constant of 17,163.31 which is also known as Y-intercept coefficient.

We can see that there is inverse relationship between the variables such that as odometer reading or age increases, car price decreases and vice versa. 

The p-values indicate statistical significance of various variables. We can see the p-values for various variables as follows:

x1 Age0.0000
x2 Odometer (kms)0.0002
x3 Transmission0.0002


In each case, the p-value is less than p = 0.05. Hence, each of these variables are statistically significant.

Further, from confidence interval data, 

x1 Age            -761.14                    -490.23 
x2 Odometer (kms)                 -0.03                         -0.01 
x3 Transmission              635.68                  1,968.14 


We can be 95% confident that:

  • With each increase in Age, the car price will decrease between -761.14 and -490.23.
  • With each increase in odometer reading (kms), the car price will decrease between -0.03 and -0.01.
  • With Transmission, the car price will change between 635.68 and 1,968.14.

Comparison

In both cases, all the variables were statistically significant. In both cases, the p-value for regression is p = 0.000 which is less than significance value of p = 0.05. Hence, both the models were statistically significant.

When we compare above two models, the parameter can be value of R2. Higher value indicates better fit and vice versa. The simple linear regression model had R2 Value of 0.69 while the multiple linear regression model had R2 Value of 0.84. 

Hence, we can say that the multiple linear regression model is a better fit as compared to simple linear regression due to higher value of R2 which indicates that almost 84% of change in car price can be explained by the three selected variables, namely age, odometer reading (kms) and transmission type.