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1. Descriptive statistics of Sale Price, Length and Weight

According to Goos &Meintrup (2015), descriptive statistics includes the measure of central tendency and measure of dispersion. The measures of central tendency are mean, median and mode, while dispersion is measured using variance, standard deviation, maximum and minimum, range, quartiles, and interquartile range. The descriptive statistics of the sales price, length and weight of the car were determined on Microsoft Excel and results are shown below.

 Statistics Sales Price Length Weight Central Tendency Mean 39699 469 1562 Median 34842 471 1545 Mode 29424 449 1716 Dispersion Variance 387164687 1000 96985 Standard Deviation 19677 32 311 Maximum 126908 557 2575 Minimum 13042 366 916 Range 113866 192 1660 Quartile(Q3) 47913 491 1733 Quartile(Q1) 26792 449 1363 Inter-quartile Range 21121 42 371

The mean is greater than the median, which is greater than the mode for the three variables. This indicates that the distributions for the three are positively skewed (Sharma 2007; Data& Using Descriptive Statistics Bartz 1988). The variances and standard deviations of the three variables are very high. Higher variance and standard is an indicator of much-dispersed data points from the mean (Bernstein& Bernstein 1998).  According to Brase& Brase (2011), a big range indicates a greater dispersion of data points, whereas a small range shows a less dispersion. Comparing the three variables, sales price has the biggest range and interquartile range, what makes its data to have the greatest dispersion among the three.

1. Estimation of a simple regression model of the Sale price on Length,

The values of   and  were determine using Microsoft Excel, regression analysis. The results are shown below.

 SUMMARY OUTPUT Regression Statistics Multiple R 0.330323 R Square 0.109113 Adjusted R Square 0.105535 Standard Error 18609.28 Observations 251 ANOVA df SS MS F Significance F Regression 1 1.06E+10 1.06E+10 30.49674 8.4E-08 Residual 249 8.62E+10 3.46E+08 Total 250 9.68E+10 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 99.0% Upper 99.0% Intercept -56711.5 17497.65 -3.24109 0.001353 -91173.8 -22249.3 -102131 -11292.6 Length 205.5067 37.2134 5.522385 8.4E-08 132.2136 278.7999 108.9112 302.1022

From the above results, the simple regression model for estimate sale price is given

• Estimation of a simple regression model of the Sale price on Length with the log-log specification.

are estimated on Excel, the results are shown below

 SUMMARY OUTPUT Regression Statistics Multiple R 0.418226 R Square 0.174913 Adjusted R Square 0.171599 Standard Error 0.177349 Observations 251 ANOVA df SS MS F Significance F Regression 1 1.660274 1.660274 52.78635 4.77E-12 Residual 249 7.831726 0.031453 Total 250 9.491999 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 99.0% Upper 99.0% Intercept -2.79362 1.011322 -2.76234 0.006167 -4.78546 -0.80178 -5.41873 -0.16851 Log Length 2.751461 0.378706 7.265421 4.77E-12 2.005585 3.497338 1.768447 3.734476

The estimated log sale price is given by

The coefficient of log length is 2.751, which is positive. According to Francis (2004) and Hassett& Stewart (2006), a positive coefficient indicates that the regression line has a positive gradient. Therefore, the estimated log sale price has a positive gradient, thus increase in length will lead to an increase in sales price.

I expected the coefficient to be a positive value above 2. The sign of the coefficient is a real representation of my expectation.

1. The Model relating the Sale price to Length and Weight;

were estimated on Excel, the results are shown below

 SUMMARY OUTPUT Regression Statistics Multiple R 0.606309658 R Square 0.367611401 Adjusted R Square 0.362511493 Standard Error 15710.28447 Observations 251 ANOVA Df SS MS F Significance F Regression 2 35581538227 1.78E+10 72.08197 2.1E-25 Residual 248 61209633442 2.47E+08 Total 250 96791171669 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 99.0% Upper 99.0% Intercept -705.1735874 15784.45647 -0.04468 0.964402 -31793.9 30383.51 -41678.4 40268.1 Length -51.78489734 40.496887 -1.27874 0.202185 -131.547 27.97679 -156.907 53.33686 Weight 41.40869765 4.112717181 10.06845 3.26E-20 33.30839 49.50901 30.73291 52.08448

The estimated sale price is given by

This model has a better goodness of fit than model in II above, its significance F, 2.1E-25, is less than that of model in II, 8.4E-08,which is less than  0.05.

1. Estimating the model in IV above using log of each variable.

The value of were estimated on Excel, results are shown below.

 SUMMARY OUTPUT Regression Statistics Multiple R 0.725841 R Square 0.526846 Adjusted R Square 0.52303 Standard Error 0.134572 Observations 251 ANOVA df SS MS F Significance F Regression 2 5.00082 2.50041 138.071 5.02E-41 Residual 248 4.49118 0.01811 Total 250 9.491999 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 99.0% Upper 99.0% Intercept 0.507216 0.804954 0.630118 0.529198 -1.0782 2.092633 -1.58228 2.596713 Log Length -0.61182 0.379339 -1.61285 0.10805 -1.35895 0.135322 -1.5965 0.372873 Log Weight 1.783179 0.131293 13.58171 8.74E-32 1.524588 2.04177 1.44237 2.123989

The estimated log sale price model is given by;

1. Testing whether length has a negative effect on sale price at 1% significance level.

Null hypothesis: Length has a negative effect on sale price.

From the above table, the P-value of Log length is 0.10805 which is greater than 0.05. This suggests that the length is not statistically significant at 1% level, the null hypothesis will be rejected (Aiken, West & Reno 1991). As a result, length does not have negative effects on the sale price.

• Adding Horsepower and luggage size to the log-log model in V.

The values of were determined on Excel, the results are shown below.

 SUMMARY OUTPUT Regression Statistics Multiple R 0.895914 R Square 0.802662 Adjusted R Square 0.799453 Standard Error 0.08726 Observations 251 ANOVA Df SS MS F Significance F Regression 4 7.618868 1.904717 250.1481848 2.04037E-85 Residual 246 1.873131 0.007614 Total 250 9.491999 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 99.0% Upper 99.0% Intercept 3.442802 0.557211 6.178627 2.66178E-09 2.345287893 4.5403157 1.99630194 4.8893017 Log Length -0.95977 0.24875 -3.85838 0.000145868 -1.449720201 -0.4698191 -1.605514 -0.31402526 Log Weight 1.041427 0.116977 8.902839 1.22434E-16 0.811022984 1.2718314 0.73775938 1.345094981 Horsepower 0.001962 0.000118 16.58606 5.58703E-42 0.001728996 0.002195 0.00165491 0.00226907 Luggage Size -0.00164 0.000582 -2.81992 0.005194904 -0.002789204 -0.0004952 -0.00315393 -0.00013042

The estimate log sale price will be

From the information in the table above, Horsepower is statistically significant at 1% level, since its P-value, 5.58703E-42 is less than 0.05. Similarly, Luggage size is significant because its P-value, 0.005194904 is also less than 0.05. The two variables are jointly significant at 5%, as 0, which is null the hypothesis is not within their 95% confidence interval brackets are above.

• The overall significance of the model in VII above at 1%.

The overall significance is determined using the significance F. The significance F, 2.04037E-85, is less than 0.05. This indicates that one of the variables is statistically significant. This means the model is good for the estimation of the sale price.

1. Testing whether Luxury cars are more expensive than other types of cars

Null hypothesis: Luxury car are not more expensive than other types of cars

 SUMMARY OUTPUT Regression Statistics Multiple R 0.846518 R Square 0.716593 Adjusted R Square 0.713151 Standard Error 0.10436 Observations 251 ANOVA Df SS MS F Significance F Regression 3 6.801904 2.267301 208.1797 2.52E-67 Residual 247 2.690096 0.010891 Total 250 9.491999 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0% Intercept 0.393903 0.624303 0.630948 0.528658 -0.83573 1.623538 -0.83573 1.623538 Log Length -0.06938 0.297186 -0.23347 0.815591 -0.65473 0.515958 -0.65473 0.515958 Log Weight 1.345826 0.107347 12.53713 3.12E-28 1.134393 1.557258 1.134393 1.557258 Luxury 0.196725 0.015298 12.85972 2.58E-29 0.166594 0.226856 0.166594 0.226856

The P- value for luxury is less than 0.05, therefore, Luxury is statistically significant at 5%, hence Luxury cars are more expensive than other types of cars.

References

Aiken, L.S., West, S.G. and Reno, R.R., 1991. Multiple regression: Testing and interpreting interactions. Sage

Bernstein, S. and Bernstein, R., 1998. Schaum's Outline of Elements of Statistics I: Descriptive Statistics and Probability. McGraw-Hill Companies.

Brase, C.H. and Brase, C.P., 2011. Understandable statistics: Concepts and methods. Cengage Learning.

Data, S. and Using Descriptive Statistics Bartz, A.E., 1988. Basic statistical concepts. New York: Macmillan. Devore, J., and Peck.

Francis, A., 2004. Business mathematics and statistics. Cengage Learning EMEA.

Goos, P. and Meintrup, D., 2015. Statistics with JMP: graphs, descriptive statistics and probability. John Wiley & Sons.

Hassett, M.J. and Stewart, D., 2006. Probability for risk management. Actex Publications

Sharma, J.K., 2007. Business statistics. Pearson Education India.

Cite This Work

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[Accessed 17 July 2024].

My Assignment Help. 'Descriptive Statistics And Regression Analysis' (My Assignment Help, 2021) <https://myassignmenthelp.com/free-samples/mae256-analytical-methods-in-economics-and-finance/descriptive-statistics-and-probability.html> accessed 17 July 2024.

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