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MAT10251  Statistical Analysis Project Assignment  Southern Cross University, Australia
Part A  Preliminary Analysis of Sample Data
a) Price of two and three year old cars
1. Construct a frequency histogram or polygon for the price of two and three year old cars.
2. Calculate descriptive statistics for the price of two and three year old cars.
b) Difference in price between cars for sale privately and those for sale by a used car dealer.
1. Construct separate boxplots, on the same plot or separately, for private sale prices and for used car dealer prices.
2. Calculate descriptive statistics for private sale prices and for used car dealer prices.
c) Relationship between price and age and between price and odometer reading
1. Construct scatter plots for Age and Price and for Odometer and Price.
2. Calculate the correlation coefficient for Age and Price and for Odometer and Price.
Solutions 
Sample Number is 5
Histogram of used Cars
Class Interval

Frequency

0  4500

6

4500  9000

19

9000  13500

17

13500  18000

20

18000  22500

14

22500  27000

29

27000  31500

8

31500  36000

1

36000  40500

1

Total

115

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Descriptive Statistics
Price



Mean

17236.57

Standard Error

759.5753

Median

16900

Mode

23990

Standard Deviation

8145.537

Sample Variance

66349781

Kurtosis

0.75397

Skewness

0.069603

Range

38190

Minimum

1800

Maximum

39990

Sum

1982205

Count

115

Difference in Price of Used Cars

Private

Dealer

Mean

14611

18466.95775

Standard Error

1104.141723

1005.02106

Median

14750

19996

Mode

15500

23990

Standard Deviation

7069.956626

8468.45798

Sample Variance

49984286.7

71714780.56

Kurtosis

0.499191626

0.829589315

Skewness

0.165449913

0.0353172

Range

28190

36991

Maximum

29990

39990

Minimum

1800

2999

Sum

599051

1311154

Count

41

71

Geometric Mean

12460.80605

16060.83378

Harmonic Mean

9603.135707

13185.61178

AAD

5641.560976

7406.494743

MAD

4750

6006

IQR

9310

12445

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Relationship between price and age
Correlation Coefficient Table

Age

Price

Age

1


Price

0.9194

1

Question  Written Answer  Preliminary Analysis  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.
Preliminary Analysis
The main objective of this study is to determine the factors that influence the price of the used cars in Australia. In the recent years, the sale of used cars has been increased drastically and there are many reasons behind it. One of the major reason is that the seller decided to buy a new car while the buyer likes to gain experience in car driving. If the buyer is new to his driving skills, there is more chance of his car being hit and get damaged. Thus, he is more interested in buying used cars which is nearly 60% less than the new car of the same make and so, he can gain experience in his driving with minimal cost involved.
For the purpose of this study, a random sample of 115 cars were taken from Queensland State and the car type of Nissan XTrail sales details was used for the analysis. Regarding the price of used cars, it is seen that about 25.2% of the used cars price ranges between $ 22500 and $ 27000, 17.4% of the used cars price ranges between $ 13500 and $ 18000 and 16.5% of the used cars price ranges between $ 4500 and $ 9000
On an average, the price of used Nissan X Trial cars is $ 17236.56 with a standard deviation of $ 837.54. The median price of used Nissan X Trial cars is $ 16900 and the recorded minimum and maximum price of used Nissan X Trial cars is $ 1800 and $ 39990 respectively.
Regarding the customer intention of buying white cars, it is seen that about 45.22% of the customers prefer white cars while 54.78% of the customers prefer nonwhite cars
On an average, the price of used Nissan X Trial cars sold privately is $ 14611 with a standard deviation of $ 7069.96. The median price of used Nissan X Trial cars sold privately is $ 14750 and the recorded minimum and maximum price of used Nissan X Trial cars is $ 1800 and $ 29990 respectively. On an average, the price of used Nissan X Trial cars sold through dealer is $ 18466.96 with a standard deviation of $ 8468.46. The median price of used Nissan X Trial cars sold through dealer is $ 19996 and the recorded minimum and maximum price of used Nissan X Trial cars is $ 2999 and $ 39990 respectively. Thus, the dealer sold used cars are sold at high price when compared with cars sold privately
Regarding the relationship between age and price of used cars, it is seen that the relationship is very strong and negative (correlation coefficient =  0.9194, p  value = 0.000 < 0.05). This indicates that as the age of the car increases, then there is a significant drop in the price of the cars
Thus, we can say that the price of the used car depends on its age and also the seller (either private or dealer) who involved in selling the car.
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Part B  Preparation
Task 1  Appendices Statistical Inference and Regression and Correlation
Question 1  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.
The descriptive statistics for the price of cars
The table given below shows the information about the price of used cars in the sample and the price of used cars sold between 2016 and 2017

Price

Price (2016  17)

Mean

17236.57

25111.08

Standard Error

759.5753

604.4957

Median

16900

24500

Mode

23990

23990

Standard Deviation

8145.537

3775.074

Sample Variance

66349781

14251186

Kurtosis

0.75397

5.310218

Skewness

0.069603

1.49746

Range

38190

22000

Minimum

1800

17990

Maximum

39990

39990

Sum

1982205

979332

Count

115

39

Here, we see that the price of used cars of the entire sample is 17236.57 ± 8145.54 and the price of used cars sold between 2016 and 2017 is 25111.08 ± 3775.07. Thus, we see that the price of used cars sold between 2016 and 2017 is high when compared with the overall sample
Null Hypothesis: H0: µ = 17236.57
The average price of used cars sold between 2016 and 2017 do not differ significantly from $ 17236.57
Alternate Hypothesis: Ha: µ ≠ 17236.57
The average price of used cars sold between 2016 and 2017 differ significantly from $ 17236.57
Level of Significance:
Let the level of significance be α = 0.05
Test Statistic
The t test statistic is
The p  value of t test statistic is 0.000. Since this p  value <0.05, we reject the null hypothesis at 5% level. Therefore, there is enough statistical evidence to conclude that average price of used cars sold between 2016 and 2017 differ significantly from the overall sample price of used cars.
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Question 2  Are more than 30% of used cars of the make and model for sale in the state specified by your sample white?
The frequency distribution of White car is given below
White

Frequency

Percentage

No

63

54.78%

Yes

52

45.22%

Total

115


Null Hypothesis: H_{0}: P <= 0.3
That is, proportion of white car purchases is not greater than 30%
Alternate Hypothesis: H_{a}: P > 0.3
That is, proportion of white car purchases is greater than 30%
Level of Significance:
Let the level of significance be α = 0.05
Test Statistic
The z test statistic
The p  value of z test statistic is 0.0002.
Since the p  value of z test statistic is less than 0.05, there is sufficient evidence to reject the null hypothesis at 5% level of significance. Thus, there is sufficient evidence to conclude that the proportion white car sales is greater than 30%.
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Question 3  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?
Null Hypothesis: H_{0}: µ_{1} = µ_{2}
That is, the average price of cars do not differ significantly between private and dealer sale
Alternate Hypothesis: H_{a}: µ_{1} ≠ µ_{2}
That is, the average price of cars differ significantly between private and dealer sale
Level of Significance:
Let the level of significance be α = 0.05
Test Statistic
The table given below shows the workings of t test statistic
tTest: TwoSample Assuming Equal Variances

Price  Private

Price  Dealer

Mean

15251.15909

18466.95775

Variance

52625931.86

71714780.56

Observations

44

71

Pooled Variance

64450882.38


Hypothesized Mean Difference

0


df

113


t Stat

2.087762093


P(T<=t) onetail

0.019533353


t Critical onetail

1.658450216


P(T<=t) twotail

0.039066706


t Critical twotail

1.981180359


The mean price of used cars sold privately is 15251.16 ± 7254.37 and the mean price of used cars sold through dealer is 18466.96 ± 8468.46. On comparing the mean values, we see that the mean price of used cars sold through dealer is significantly high when compared with the cars sold privately (t test statistic =  2.088, p  value = 0.039 < 0.05)
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Question 4  Simple Linear Regression Model  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?
Dependent Variable: Price
Independent Variable: Age
The regression output is given below
SUMMARY OUTPUT



Regression Statistics

Multiple R

0.88

R Square

0.78

Adjusted R Square

0.77

Standard Error

3316.00

Observations

115.00

ANOVA


df

SS

MS

F

Significance F


Regression

1.00

4307212746.58

4307212746.58

391.71

0.00


Residual

113.00

1242534410.41

10995879.74




Total

114.00

5549747156.99













Coefficients

Standard Error

t Stat

Pvalue

Lower 95%

Upper 95%

Intercept

25292.72

554.91

45.58

0.00

24193.34

26392.11

Age

1390.91

70.28

19.79

0.00

1530.14

1251.68

The regression equation is
Price = 25292.72  1390.91 * Age
Coefficient of Determination = 0.78
Dependent variable variation explained from the model = 78%
Unexplained variation = 22%
Here, we see that the model is good fit in predicting the price of used cars using Age as independent variable. Regarding the relationship between age and price of used cars, it is seen that the relationship is very strong and negative (correlation coefficient =  0.9194, p  value = 0.000 < 0.05). This indicates that as the age of the car increases, then there is a significant drop in the price of the cars.
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Question 5  Multiple Linear Regression Model  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.
Dependent Variable: Price
Independent Variable: Age, Odometer (kms) and Transmission
The multiple regression output is given below
SUMMARY OUTPUT

Regression Statistics

Multiple R

0.882

R Square

0.778

Adjusted R Square

0.772

Standard Error

3333.440

Observations

115.000

ANOVA


df

SS

MS

F

Significance F


Regression

3.000

4316335014.541

1438778338.180

129.482

0.000


Residual

111.000

1233412142.451

11111821.103




Total

114.000

5549747156.991













Coefficients

Standard Error

t Stat

Pvalue

Lower 95%

Upper 95%

Intercept

24702.362

858.535

28.773

0.000

23001.117

26403.608

Age

1362.512

141.809

9.608

0.000

1643.517

1081.507

Odometer (kms)

0.000

0.008

0.009

0.993

0.015

0.015

Transmission

639.346

705.687

0.906

0.367

759.019

2037.712

The regression equation is
Price = 24702.36  1362.51 * Age  0.0001 * Odometer + 639.35 * Transmission
The coefficient of determination is 0.778, indicating that 77.8% of the variation in price is explained by the regression model and the remaining 22.2% left unexplained.
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On comparing the simple linear regression and multiple linear regression models, we see that there is no much difference in the coefficient of determination between these two models. Thus, we can conclude that the simple linear regression model is the best as it explains more variation of price with single independent variable when compared to that of multiple linear regression models.
The t test for significance of slope was used to test whether the independent variable is a significant predictor of dependent variable price. From the above regression output, we see that
1. Age is a significant predictor of price of used cars (t test statistic =  9.61, p  value < 0.05).
2. Odometer is insignificant predictor of price of used cars (t test statistic = 0.009, p  value 0.993 > 0.05).
3. Transmission  Automatic is insignificant predictor of price of used cars (t test statistic = 0.906, p  value 0.367 > 0.05).
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