Business Statistics Nonparametric Models

Question: Define the Business Statistics for Nonparametric Models . Answer: The level of measurement of the variable Size is nominal. This variable is measured in nominal scale as the values of the variable is differentiated based on the various sizes of the compact, midsize and large (Pedhazur and Schmelkin, 2013). The level of measurement of displacement is ratio scale as the values of the variable are measured in numbers. The level of measurement of cylinders is ratio scale. This is because the values of the variable give the magnitude of the cylinders of different sizes. The level of measurement of drive is ordinal. This is because the variable is ordered according to their wheels. The variable is classified according to all wheel, front wheel and rear wheel which denotes the order of the variable. The level of measurement of the variable fuel type is ordinal (Gries, 2014). This is because the values of this variable are classified according to the premium fuel or regular fuel. The variable city MPG has the level of measurement as ratio scale. This is because the values of the variable have a particular magnitude that gives the fuel efficiency rating for the city driving in terms of miles per gallon. The level of measurement of the variable Hwy MPG is ratio scale. This is because the there is a particular value for the samples of this variable which gives the measure of fuel efficiency rating for highway driving in terms of miles per gallon. Refer to the excel file in the excel sheet Data MG14. The histogram of cylinders is given below: Figure 1: Histogram of the variable cylinders (Source: Created by author) The histogram of the variable cylinders show that the minimum value of the variable is 4 and the maximum value of the variable is 12. The frequency of maximum value of the variable cylinder is less. This shows that there are few cars that have maximum number of engines are 12. Most of the cars use 4, 6 or 8 engines. The histogram shows that the distribution of the variable is not normal, as the curve of the variable do not follow the curve of normal distribution. Refer to the excel sheet question f. The table of relative frequencies and percent frequencies for the frequency distribution of the variable cylinders is given below: values relative frequency 4 = (48 / 100) = 0.48 5 = (4 / 100) = 0.04 6 = (27 / 100) = 0.27 8 = (18 / 100) = 0.18 12 = (3 / 100) = 0.03 Table 1: Table of relative frequency of the variable cylinders (Source: created by author) values percentage frequency 4 = (48/100) * 100 = 48 5 = (4/ 100) * 100 = 4 6 = (27 / 100) * 100 = 27 8 = (18 / 100) * 100 = 18 12 = (3 / 100) * 100 = 3 Table 2: Table of percentage frequency of the variable Cylinders (Source: Created by author) Refer to the excel sheet question h. The pivot table constructed in excel is given below: Count of Cylinders Column Labels Row Labels 4 5 6 8 12 Grand Total Compact 24 4 6 3 1 38 Large 3 4 12 1 20 Midsize 21 17 3 1 42 Grand Total 48 4 27 18 3 100 Table 3: Pivot table using size as row label, cylinders as column label and count of cylinders as the values of the pivot table (Source: created by author) The probabilities are calculated using the pivot table created in the variable cylinders: Total number of cylinders is 100. Number of cars that have 4 cylinders is 48. The probability of selecting 4 cylinders at random is given by (48 / 100) = 0.48. It was seen that the total number of cars is 100. Number of cars whose size is compact; i.e. small size is 38. The probability of randomly selecting a car, which has small size is given by (38 / 100) = 0.38. The total number of cars chosen as samples is 100. The number of cars, which are small, is 38 and the number of cars which have 4 engines is 48. The number of small cars who have 4 engines is 24. The probability of selecting small cars which have 4 engines is given by 24 / 100 = 0.24. The descriptive statistics calculated for the variable City MPG is given below: City MPG Mean 19.96 Standard Error 0.468593425 Median 19 Mode 18 Standard Deviation 4.68593425 Sample Variance 21.9579798 Kurtosis 3.204308984 Skewness 1.073940138 Range 30 Minimum 11 Maximum 41 Sum 1996 Count 100 Largest(1) 41 Smallest(1) 11 Confidence Level (95.0%) 0.929790993 Table 4: descriptive statistics of City MPG (Source: created by author) The descriptive statistics calculated for the variable Hwy MPG is given below: Hwy MPG Mean 28.93 Standard Error 0.520538765 Median 29 Mode 29 Standard Deviation 5.205387652 Sample Variance 27.09606061 Kurtosis 0.059249286 Skewness 0.163426126 Range 24 Minimum 18 Maximum 42 Sum 2893 Count 100 Largest(1) 42 Smallest(1) 18 Confidence Level (95.0%) 1.032861815 Table 5.: descriptive statistics of Hwy MPG (Source: created by author) On performing descriptive statistics on the variable City MPG, the mean value was found to be 19.96 while the value of standard deviation was found to be 4.68. It can be interpreted that the mean fuel efficiency rating for the city driving in miles per gallon is 19.96. The value of standard deviation is medium. It can be interpreted that the spread of fuel efficiency rating for city driving is deviated moderately from its mean value. The descriptive statistics of the variable Hwy MPG shows that the mean value of the variable was 28.93 and the standard deviation of the variable was 5.20. It can be interpreted that the average rating of fuel efficiency for driving on highway in terms of miles per gallon is 28.93 (Weiss and Weiss, 2012). The rating shows that the efficiency of the fuel for driving on highways is high as the average value is high. The value of standard deviation shows that the efficiency of fuels deviates moderately from the mean with a value of 5.20. The deviation of the variable shows that the efficiency of fuels for driving on highway variers moderately across the cars. The sample of City MPG and Hwy MPG drawn from its population is not given to follow normal distribution. Thus, it is assumed that the distribution of the population is t-distribution where the standard error of the population would be estimated from the sample drawn from the population and it would be used instead of standard deviation. In order to calculate the margin of error for 95% confidence interval for the mean of the population of the variable City MPG, the standard error of the variable is found by (standard deviation / sqrt (n) = 4.68 / 10 = 0.468. The 95% critical value of this variable following t-distribution and having degree of freedom as 99 is given as 1.66. Therefore, the margin of error of City MPG at 95% confidence interval when the variable follows t-distribution is given by 1.66 * 0.468 = 0.7769. In order to calculate the margin of error for 95% confidence interval for the mean of the population of the variable Hwy MPG, the standard error of the variable is found by (standard deviation / sqrt (n) = 5.20/ 10 = 0.520. The variable follows t-distribution and has the degrees of freedom as 99 (Bickel and Lehmann, 2012). The value of 95% confidence interval having 99 degrees of freedom is 1.66 (Kock, 2013). The margin of error for the variable Hwy MPG is 1.66 * 0.52 = 0.8632. The 95% confidence interval for population mean of the variable City MPG is given by mean +(-) 1.66* standard error (Huang and Bentler, 2015). The lower 95% interval is given by 19.96 (1.66 * 0.468) = 19.183 and the upper 95% interval is given by 19.96 + (1.66 * 0.468) = 20.736. The 95% confidence interval is given by 20.736 19.183 = 1.554. It can be interpreted that the 0.95 probability of containing the population mean is 1.554. The 95% confidence interval for population mean of the variable Hwy MPG is given by mean +(-) 1.66* standard error. The lower 95% interval is given by 28.93 (1.66*0.520) = 28.067 and the upper 95% interval is given by 28.93 + (1.66*0.520) = 29.793. The 95% confidence interval of Hwy MPG is given by 29.793 28.067 = 1.726. It can be interpreted that 0.95 probability of containing the population mean for this variable is 1.726. 0854 give the covariance between the variable Displacement and City MPG. The correlation between the variable Displacement and City MPG is given by -0.72805. 0837 give the covariance between the variable Displacement and Hwy MPG. The correlation between the variable Displacement and Hwy MPG is given by -0.81555. The correlation coefficient between Displacement and City MPG was found to be -0.72805. It is seen that there is a strong negative relationship between the two variables. It can be interpreted that the change in one variable would have a strong effect on the other variable but in the opposite direction (Sang et al., 2016). This suggests that more is the value of Displacement less is the value of City MPG. The correlation between Displacement and Hwy MPG was found to be -0.81555, which defines a strong negative association between these two variables. It can be interpreted that the change in one variable would strongly influence the change in another variable in the opposite direction (Shu and Nan, 2014). This suggests that higher the change in Displacement, lower is the value of Hwy MPG. References Bickel, P.J. and Lehmann, E.L., 2012. Descriptive statistics for nonparametric models IV. Spread. InSelected Works of EL Lehmann(pp. 519-526). Springer US. Gries, S.T., 2014. Frequency tables: tests, effect sizes, and explorations.Glynn D, Robinson J. Polysemy and synonymy: corpus methods and applications in cognitive linguistics. Amsterdam: John Benjamins. Huang, Y. and Bentler, P.M., 2015. Behavior of asymptotically distribution free test statistics in covariance versus correlation structure analysis.Structural Equation Modeling: A Multidisciplinary Journal,22(4), pp.489-503. Kock, N., 2013. Using WarpPLS in E-Collaboration Studies: Descriptive Statistics, Settings.Interdisciplinary Applications of Electronic Collaboration Approaches and Technologies,62. Pedhazur, E.J. and Schmelkin, L.P., 2013.Measurement, design, and analysis: An integrated approach. Psychology Press. Sang, Y., Dang, X. and Sang, H., 2016. Symmetric Gini Covariance and Correlation.arXiv preprint arXiv:1605.02332. Shu, H. and Nan, B., 2014. Large covariance/correlation matrix estimation for temporal data.arXiv preprint arXiv:1412.5059. Weiss, N.A. and Weiss, C.A., 2012.Introductory statistics. London: Pearson Education.