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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 SAVE YOUR HIGHER GRADE WITH ACQUIRING STATISTICAL ANALYSIS PROJECT ASSIGNMENT HELP & QUALITY HOMEWORK WRITING SERVICES OF EXPERTSMINDS.COM!

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 X-Trail 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 non-white 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: H0: P <= 0.3

That is, proportion of white car purchases is not greater than 30%

Alternate Hypothesis: Ha: 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: H0: µ1 = µ2

That is, the average price of cars do not differ significantly between private and dealer sale

Alternate Hypothesis: Ha: µ1 ≠ µ2

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

t-Test: Two-Sample 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) one-tail 0.019533353 t Critical one-tail 1.658450216 P(T<=t) two-tail 0.039066706 t Critical two-tail 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 P-value 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 P-value 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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