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    Statistics and Research Method for Business Decision Making

    Question 1

    a) Appropriate Graphical Technique for Comparing the Amount of CO2 Emissions in the year 2009 and 2013

     

    Here, bar diagram has been used to compare the amount of carbon die oxide emissions in the year 2009 and 2013. It is obtained from the graph that, the amount of CO2 emission was higher in China for both the years, in the year 2013 the amount reaches its maximum. The data shows that, Australia faced the lowest amount of CO2 emission in both the years that is 2009 and 2013. Furthermore, it is noted from the data that, the amount of CO2 emission reduces over the year in almost every country.

    b) Appropriate Graphical Technique for Comparing the Percentage Value of the Amount of CO2 Emissions in 2009 and 2013

     

     

     

    The percentage value of the amount of CO2 emissions in 2019 and 2013 has been drawn with the help of a line diagram. It is observed that, CO2 emissions percentage reaches its maximum in the year 2013. However, the percentage of CO2 emissions reduces over the year in all the countries mentioned.

    c) Here, a comparison of the amount of CO2 emission has been observed for United States, China, Russia, Japan, India, Germany, Canada, United Kingdom, South Korea, Italy, Iran, South Africa, France, Saudi Arabia and Australia. From the above two diagram, it is noted that the amount of CO2 emission decreases from the 2009 to 2013 in all most every countries. However, in case of percentage of CO2 emissions also decreases in almost every country from the year 2009 to the year 2013.

     

     

     

     

     

    Question 2

    Classes

    Frequency

    Relative Frequency

    Cumulative Frequency

    Cumulative Relative Frequency

    35-44

    3

    0.075

    3

    0.075

    45-54

    4

    0.1

    7

    0.175

    55-64

    9

    0.225

    16

    0.4

    65-74

    18

    0.45

    34

    0.85

    75-84

    4

    0.1

    38

    0.95

    85-94

    1

    0.025

    39

    0.975

    95-104

    1

    0.025

    40

    1

     

    a) Appropriate Frequency Distribution

    Frequency Distribution, frequency distribution table helps to identify the frequencies of various outcomes of a sample. More precisely, frequency table summarizes all the distribution values included in a sample (Piepho, 2017). It is a useful technique as it helps researcher to understand the occurrence of mean within a dataset. Here, the last value should always be the sum of total observations. Relative frequency can be obtained by dividing the frequency by the sum of the total observations.

    Relative Frequency= Frequency / Total Frequency

    Classes

    Frequency

    Cumulative Frequency

    35-44

    3

    3

    45-54

    4

    7

    55-64

    9

    16

    65-74

    18

    34

    75-84

    4

    38

    85-94

    1

    39

    95-104

    1

    40

     

    b) Appropriate Cumulative Frequency Distribution 

    Cumulative frequency distribution is a type of frequency distribution; it is the sum of classes (Nakagawa et al., 2017). This cumulative frequency can be calculated by adding every frequency present in a frequency distribution. Cumulative relative frequency can be obtained by dividing cumulative frequency for one observation by total of cumulative frequency.

    Cumulative Frequency = F1 + (F1+ F2) + (F1+ F2 + F3) and so on.

    Relative Cumulative Frequency = Cumulative Frequency / Total Cumulative Frequency

    Classes

    Frequency

    Cumulative Frequency

    Cumulative Relative Frequency

    35-44

    3

    3

    0.075

    45-54

    4

    7

    0.175

    55-64

    9

    16

    0.4

    65-74

    18

    34

    0.85

    75-84

    4

    38

    0.95

    85-94

    1

    39

    0.974

    95-104

    1

    40

    1

     

    c) Relative Frequency Histogram

     

    Relative frequency histogram is one type of graph that shows the percentage of the frequencies. Here, the data shows that the amount of time required for assembly line workers so that they can complete a weld at a car assembly plant and the number of worker here is 40. It is obtained from the data that, relative frequency histogram is highest in the class 65 to 74, that is 0.45.

    d) Ogive

     

    Ogive graph plots cumulative frequency on the horizontal axis that is y axis and class boundaries along with the vertical axis. An ogive is a graph that plots the cumulative frequencies and allows estimating number of observations (Piepho, 2018). Here, cumulative frequency has been plotted in the horizontal axis.

    e) The Proportion of Data Less than 65

    The proportion of data less than 65 is 15%. It has been obtained from the above table.

    f) The Proportion of Data More than 75

    The proportion of data more than 75 is 66%, it has been obtained from the above table.

     

     

    Question 3

    SUMMARY OUTPUT

                 
                     

    Regression Statistics

                 

    Multiple R

    0.038875

                 

    R Square

    0.001511

                 

    Adjusted R Square

    -0.05104

                 

    Standard Error

    1268.43

                 

    Observations

    21

                 
                     

    ANOVA

                   

     

    df

    SS

    MS

    F

    Significance F

         

    Regression

    1

    46268.68

    46268.68

    0.028758

    0.867132

         

    Residual

    19

    30569396

    1608916

             

    Total

    20

    30615664

     

     

     

         
                     

     

    Coefficients

    Standard Error

    t Stat

    P-value

    Lower 95%

    Upper 95%

    Lower 95.0%

    Upper 95.0%

    Intercept

    3874.286

    696.832

    5.559857

    2.31E-05

    2415.8

    5332.773

    2415.8

    5332.773

    Rate of Inflation

    40.30771

    237.6901

    0.169581

    0.867132

    -457.183

    537.7989

    -457.183

    537.7989

     

    1. Descriptive Measures (Time Series Data)

     

    For understanding the graphical measures used for time series data a simple line diagram has been used. Line diagram is the best suited for understanding the trend line of time series data. Here, two variables that are rate of inflation and all ordinaries index has been drawn with the help of line diagram. It is observed from the data that, the trend line of rate of inflation is upward rising and it increases over the years. Whereas, the variable all ordinaries index is follows a straight line trend that is horizontal to the y axis. 

    b) Scatter Plot

     

    It is observed from the data that, there exist no trend line between the variables rate of inflation and all ordinaries index. Scatter diagram is a set of points that significantly portrayals the relationship between rate of inflation and all ordinaries index (Barnes and Barnes, 2015). It can be stated that there is very poor correlation between rate of inflation and all ordinaries index. Here, all ordinaries index has been plotted to horizontal axis and rate of inflation has been plotted to x-axis.

    c) Numerical Summary Report 

    Rate of Inflation

     

    All-Ordinaries Index

     

           

    Mean

    2.690476

    Mean

    3982.733

    Standard Error

    0.260394

    Standard Error

    269.9897

    Median

    2.5

    Median

    4127.6

    Mode

    2.4

    Mode

    #N/A

    Standard Deviation

    1.193275

    Standard Deviation

    1237.248

    Sample Variance

    1.423905

    Sample Variance

    1530783

    Kurtosis

    2.038282

    Kurtosis

    -1.01001

    Skewness

    0.798408

    Skewness

    0.182747

    Range

    5.6

    Range

    4336.8

    Minimum

    0.3

    Minimum

    2000.8

    Maximum

    5.9

    Maximum

    6337.6

    Sum

    56.5

    Sum

    83637.4

    Count

    21

    Count

    21

     

    The summary report includes summary measures of rate of inflation and all ordinaries index. It is obtained from the summary measures that, the mean, median, range, variance and standard deviation of rate of inflation are 2.690476, 2.5, 5.6, 1.423905 and 1.193275 respectively. On the other hand, the mean, median, range, variance and standard deviation of all ordinaries index are 3982.733, 4127.6, 4336.8, 1237.248 and 1530783 respectively. The three quartiles of rate of inflation are first quartile (Q1) = 2.4, second quartile (Q3) = 2.5 and third quartile (Q3) = 2.9. The first quartile (Q1) of all ordinaries index is 3032, second quartile (Q2) of all ordinaries index is 4127.6 and third quartile (Q3) of ordinaries index is 4933.5.

    Coefficient of Variation = (Standard Deviation / Mean)*100

    The value of coefficient of variation (COV) for rate of inflation is 44.35180694 and for all ordinaries index is 31.0653047.

    d) Correlation Coefficient

     

    Rate of Inflation

    All-Ordinaries Index

    Rate of Inflation

    1

     

    All-Ordinaries Index

    0.038875

    1

     

    Coefficient of Correlation (r ) between rate of inflation and all ordinaries index is 1. This correlation shows the relation between the variables rate of inflation and all ordinaries index. Thus, it can be stated that there is a strong relationship between both the variables rate of inflation and all ordinaries index.

    e) Estimating Regression Line

    Simple Linear Regression Equation is,

    All Ordinaries Index = 3874.286 + 40.30771 (Rate of Inflation)

    Here, all ordinaries index is the dependent variable and rate of inflation is the independent variable. It is obtained from the regression equation that there is a strong positive correlation between all ordinaries index and rate of inflation. 

    f) Estimating Coefficient of Determination

    The coefficient of Determination is the R2 value; it is obtained from the ANOVA table of the regression analysis. It is obtained from the data that the value of R2 is 0 this indicates that rate of all ordinaries index that is the dependent variable cannot be predicted with the help of all ordinaries index (independent variable).

    g) Testing of Significance

    Null Hypothesis (H0): All ordinaries index has no positive relation with the rate of inflation.

    Alterative Hypothesis (H1): All ordinaries index has a positive relation with the rate of inflation

    Testing of significance can be done with the help of p value; p value helps to determine the significance of the entire result. P value comes in between the numbers 0 and 1, if the value is less than 0.05, then test result would not be significant and if the value is greater than 0.05, then result would be significant (Altman and Krzywinski, 2015). In this case, the p value obtained is 0.01 that is lower than the value 0.05, therefore, it can be stated that both the variables are not significant.

    h) Standard Error

    Standard error of estimate measures the variation of a variable that can been occurred around the computed regression line. Variable prediction accuracy can be measured with the help of standard error of estimates that has been generated from the regression line. However, the value of standard error of estimate obtained here is 1.22. This value indicates that the fitness of linear regression model is high and both variables are significantly related.

    References

    Altman, N. and Krzywinski, M., 2015. Points of significance: simple linear regression.

    Barnes, E.A. and Barnes, R.J., 2015. Estimating linear trends: Simple linear regression versus epoch differences. Journal of Climate28(24), pp.9969-9976.

    Halsey, L.G., Curran-Everett, D., Vowler, S.L. and Drummond, G.B., 2015. The fickle P value generates irreproducible results. Nature methods12(3), p.179.

    Nakagawa, S., Johnson, P.C. and Schielzeth, H., 2017. The coefficient of determination R 2 and intra-class correlation coefficient from generalized linear mixed-effects models revisited and expanded. Journal of the Royal Society Interface14(134), p.20170213.

    Piepho, H.P., 2018. A Coefficient of Determination (R2) for Linear Mixed Models. arXiv preprint arXiv:1805.01124.

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