Statistical Methods in Business & Economics-myassignmenthelp

Question: Discuss about theStatistical Methods in Business Economics for Fit Line. Answer: The objective is to ascertain the best estimate of correlation coefficient from the given scatter plot. A correlation coefficient has two aspects namely the sign and the magnitude. The sign of the scatter plot is evident from the slope of the best fit line. If there is a positive slope of the best fit line or it is sloping upwards, then the correlation coefficient would be positive. Otherwise, if there is a negative slope of the best fit line or it is sloping downwards, then the correlation coefficient would be negative (Eriksson Kovalainen, 2015). Further, to determine the magnitude, it is apparent to consider that the correlation coefficient represents the linear relationship between the given variables. Thus, the higher the linearity in the distribution of the points on the scatter plot, higher would be the magnitude of the correlation coefficient. Also, it is noteworthy that the magnitude of the correlation coefficient always lies between 0 and 1 (Flick, 2015). Based on the above theoretical discussion, the correlation coefficient can be estimated. The first observation is that the best fit line has a negative slope. This is apparent from the fact that as the value on the X axis tends to increase, there is a corresponding decrease in the value of Y axis. This clearly highlights the fact that there exists an inverse relationship between the two. As a result, the change in the dependent variable is opposite to the change in the independent variable (Hair et. al., 2015). Thus, it would be appropriate to conclude the sign of the correlation coefficient would be negative. Having determined the sign of the correlation coefficient, the next task is to estimate the magnitude. For this, a best fit line needs to be drawn for the given points and the deviations of the various points needs to be observed (Harmon, 2011). It is noticeable that the best fit line is one which tends to minimise the sum of square of deviations of the observations. Clearly, it is evident that most of the points on the scatterplot are adhering to the inverse relationship between the two variables. Also, the deviations from the best fit line would not be small only which is representative of the correlation between the given variables to be strong. However, since the trend is not absolutely linear, hence, the correlation coefficient magnitude would not cross 0.9 but also would not lie below 0.7 owing to the high linearity visible in the given scatter plot. Hence, in order to narrow down on a best estimate of correlation coefficient, it would be appropriate to take the average value of the proposed interval of correlation estimate. This would come out as 0.8. Hence, based on the above discussion, it is apparent that the best estimate of the correlation coefficient based on the given scatter plot can be taken as -0.8. It is no teworthy that in the absence of the exact scale, the above value is a best estimate and the actual value may deviate but only slightly (Hastie, Tibshirani Friedman, 2011). Time series data are quite common in various fields particularly macroeconomics where economists and other researchers often need to analyse the historical data in order to decipher trends or to make sense of the present data on the various indicators. The index numbers tend to capture the fluctuations in price or quantity keeping a particular time or year as the base or reference point. Hence, using the index numbers, it is possible to measure the change both on an yearly basis and also from the reference or base year. The use of index numbers tends to make computations convenient and more importantly easy to understand which makes various patterns evident (Koch, 2013). One of the key usages of the concept of index numbers is to measure various economic indicators particularly inflation. The base year is usually given an index value of 100 and the index values are computed for subsequent time periods so as to comment on the inflation witnessed during the given period. This method is quite convenient as the there is a fixed basket of goods and services using the price of which the index value at various time intervals can be computed. Then the percentage change can be measured either from the beginning or from the period observations period. Using this, the economists can thus comment on the overall inflation trend and policy making can be done keeping the inflation trend in mind (Lehman n Romano, 2006). In the absence of the index numbers, it would be quite complicated to compute the inflation. This can be highlighted using the following example. Let us assume that CPI index for a given country X was fixed at 100 in 2012. The corresponding value of the CPI index for 2013 came out as 103.8. This implies that the annual inflation level would be 3.8% p.a. Annual inflation for 2013 = [(103.8-100)/100]*100 = 3.8% Let us go forward and assume that the CPI index for the next year i.e. 2014 comes out as 105.8 Then, annual inflation for 2014 = [(105.8-103.8)/103.8]*100 = 1.92% Similarly the inflation for the various time periods can be easily computed and the trend analysed. It is noteworthy that there are various methods of computing index numbers. One of these is the weighted average methods whereby weights are given to various articles for the computation of the index number. While this method makes sense but it has certain shortcomings. One of these is that due to changing basket of goods, the index numbers may not be comparable. Also, the determination of appropriate weights for the different items can be an issue. To avoid this, the equal weights or unweighted method may be used to construct the index (Lieberman et. al., 2013). However, for measuring economic indicators like inflation it would be unwise to place equal emphasis on all articles and hence weighted index method would be preferred. Despite various advantages of index numbers, one shortcoming is that with the changing preferences, the composition would need to be altered or accuracy would be adversely impacted (Koch, 2012). One of the key roles of statistical analysis is to conduct testing on the sample data so as to yield useful information about the population. Depending upon the number of samples, sample size and underlying distribution, different statistical tests may be applied. These are discussed below (Medhi, 2001). One sample mean test This test is used to infer whether the population mean is different (two tail), greater or lower (one tail) than a given value. The test statistic may be t or z based on the underlying sample size and also whether the population standard deviation is known or not. Further, the test is conducted based on two approaches namely the p value approach and also the critical value approach. Two sample mean test The rest is used to test whether the mean of the two samples. This could be done using either t or z test statistic. The test could be single tailed or two tailed depending upon the alternative hypothesis sign. Further, the test is conducted based on two approaches namely the p value approach and also the critical value approach. The two samples may be dependent or independent based on which the underlying approach would be altered. ANOVA test In order to compare the means of more than two samples, the ANOVA test is used. Here, the T statistic and the Z statistic cannot be used. Instead, ANOVA test leads to significance F value and the hypothesis testing is performed by comparing the significance F with the significance level. One of the assumptions for the ANOVA test is that the variances of the samples are the same. Chi Square Test At times, it needs to be tested whether the relationship between the given set of variables is significant or not. For this, the chi square statistic needs to be computed. Further, the hypothesis testing can be performed using either p value approach or critical value approach. The formula for chi square test statistic computation is based on the difference between observed and expected values. F test for variances In order to test whether the variances of two populations are different or not, the F test may be conducted on the given two samples. Further the p value approach or critical value approach may be used to decide whether the null hypothesis would be rejected or not. Some examples of the above tests are as follows. A fictitious output of the two sample mean testing using t statistic is indicated below. Assuming a single tail test, the p value would be 0.00 and since the p value is lesser than level of significance (assumed 1%), hence the null hypothesis would be rejected. For a two tail test also, the p value is lesser than level of significance (assumed 1%), hence the null hypothesis would be rejected. The above conclusion can also be reached using the critical value approach (Lind, Marchal Wathen, 2012). References Eriksson, P. Kovalainen, A. (2015).Quantitative methods in business research (3rd ed.). London: Sage Publications. Flick, U. (2015).Introducing research methodology: A beginner's guide to doing a research project (4th ed.). New York: Sage Publications. Hair, J. F., Wolfinbarger, M., Money, A. H., Samouel, P., Page, M. J. (2015).Essentials of business research methods (2nd ed.). New York: Routledge. Harmon, M. (2011). Hypothesis Testing in Excel - The Excel Statistical Master (7th ed.). Florida: Mark Harmon. Hastie, T., Tibshirani, R. Friedman, J. (2011).The Elements of Statistical Learning (4th ed.).New York: Springer Publications. Hillier, F. (2006). Introduction to Operations Research. (6th ed.). New York: McGraw Hill Publications. Koch, K.R. (2013). Parameter Estimation and Hypothesis Testing in Linear Models (2nd ed.). London: Springer Science Business Media. Lehman, L. E. Romano, P. J. (2006). Testing Statistical Hypotheses (3rd ed.). Berlin : Springer Science Business Media. Lieberman, F. J., Nag, B., Hiller, F.S. Basu, P. (2013). Introduction To Operations Research (5th ed.). New Delhi: Tata McGraw Hill Publishers. Lind, A.D., Marchal, G.W. Wathen, A.S. (2012). Statistical Techniques in Business and Economics (15th ed.). New York : McGraw-Hill/Irwin. Medhi, J. (2001). Statistical Methods: An Introductory Text (4th ed.). Sydney: New Age International.