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Question:

1:

Pawn broking business started in the early 19th century in Singapore. By the end of the 21st century, it serves the "white collar" customers in the country. Pacific Pawn Brokers is one of the leading pawn broking company, that operates in Hong Kong. The firm wants to start a business in Singapore. The company intends to be among the top three businesses in the country. So the company wishes to know about the state of the economy of the country for running this business. This requires statistical investigation of previous years data relating to the number of pledges received, the number of pledges redeemed; some loans are given, etc.

A sample of monthly pawnshop data from the year 1980 to the year 2015 has been taken from the site of Department of Statistics, Statistical Tables from singstat Table Builder for the analysis (Tablebuilder.singstat.gov.sg, 2016).The figures of a monthly number of pledges received per month have been segmented into two parts - before the year 2000 and after the year 2000. Various descriptive statistics measures have been used to figure out the change in some pledges received.

The data is basically a time series data. Plotting the values of No. Of pledges against the dates the above graph is obtained. The graph shows a steady increase in the costs from 1980 to May 2010 and a rapid growth in the values after that. No such outliers can be spotted from the graph.

The measures of central tendency used are arithmetic mean and median (Boone and Boone 2012).

The mean and the median values before the year 2000 has been 195988 and 194178 respectively. The costs for the years after 2000 are 264511 and 242501. There is a great change in central tendency values before and after 2000. The central tendency value reveals that the number of pledges after the year 2000 has increased more than 25%.

The measures of dispersion used in this case are the range, inter quartile range and standard deviation (Bickel and Lehmann 2012).

90% of the values of a variable is expected to lie within the inter quartile range. A significant difference between the values of range and Inter quartile range indicates the presence of outlier ( Wan et al. 2014).

Before the year 2000, the values of range and inter quartile range are 126519 and 41658.0 respectively while that after 2000 is 176498 and 88168 respectively. The difference between the range and inter quartile range is more for the years after 2000 than for the years before 2000. The standard deviation values for the number of pledges received before the year 2000 is 26975.98 while that after the year 2000 is 52275.27

The measures of central tendency and measures of dispersion values reveal that the number of pledges received after the year 2000 is rapidly growing. The values after 2000 are greatly dispersed and has a heavy-tailed distribution. The result means the numbers of pledge received is increasing very fast in the current years. This is also evident from the graph (KoÅ‚acz and Grzegorzewski  2016).

2:

The mean value of number of pledges after the year 2000 is 264511.The probability that the sample mean is above 26000 has to be calculated. The likelihood of the value is computed to ensure if the sample is repeated will the value be also higher than 26000.The sample size is 188.As the scale of the sample is considerably large, (more than 30), the population standard deviation can be approximated by sample standard deviation. The distribution of the sample mean can be assumed to be a standard normal (Hoenig and Heisey 2012).

So the probability that the sample mean is greater than 260000 is given by:

=1-Ñ„(1.18319)

=1-0.19886

=0.80114.

The upper and lower confidence limit is given by:

UCL=Xbar +s/sqrt(n)*z

LCL= Xbar-s/sqrt(n)*z

The tabulated value of z at 90% level of significance is given by 1.282(Altman et al. 2013).

So the confidence interval for pledge received after the year 2000 is:

UCL=264511+3812.566*1.282=271199.205

LCL=264511-3812.566*1.282=261270.59425804

The confidence interval for pledged received before the year 2000 is:

UCL=195988+1.282*1741.292=198159.483654

LCL=195988-1.282*1741.292=193688.516346

The confidence interval for the period before the year 2000 is (193424.667926,1948423.332074) and the confidence interval for the period after the year 2000 is (271199.25,261270.59425804). This means that if the sample is repeated as many times as required the value of the average will lie within this interval. So it can be concluded that the mean value for the period of the year 2000 is greater than the mean value before the year 2000.

C.The assumptions made for the purpose of constructing the confidence interval are:

1.The population standard deviation has been approximated by sample standard deviation as the sample size is quite large.

2.The distribution of sample mean is assumed to be normal.

If the sample standard deviation cannot be approximated by population standard deviation, then estimated value of sample standard deviation is to be used. Then the distribution of test statistic will be t distribution instead of standard normal(Aron, Coups and Aron 2013).

The confidence interval will be:

UCL= Xbar +s/sqrt(n)*t

LCL= Xbar -s/sqrt(n)*t

The value of t at 90% confidence limit for df = 191 is 1.660.

Then the confidence interval will be (193097.5,198878.6) for the period before 2000 and (258182.2,270839.9) for the period after 2000(Kruschke  2013).

3:

There has been a financial crisis in the year 2008-2009 which has made the director of the Pacific Pawn Brokers company to think that the amount of loans redeemed including interest has been lowered after this financial crisis. To verify the statement a test has been conducted to check whether the mean amount of loans redeemed before and after the financial crisis has changed significantly or not.

Here µ2 represents the mean amount of loans redeemed before 2009 and µ1 represent the mean amount of loan redeemed after the year 2009.So to test whether the two mean values are equal, or the mean value has been increased after the financial crisis (2008-2009) is to test H0:  against  H1: µ1>µ2(Bera, Galvao and Wang  2014).

The value of population standard deviation has to be estimated from sample standard deviation. The test statistic for the purpose of testing is given by:

T=

Where sp denotes the square root of the pooled variance.

Where s1 is the standard deviation of the first sample and s2 is the standard deviation of the second sample. The value of pooled variance is 136.8934.

The test statistic is said to follow a t distribution with (n1+n2-2)=182 degrees of freedom. Under the null hypothesis, the value of the t statistic is 63.09618.

The tabulated value of t statistic at 0.05% level of significance for degrees of freedom= 182 is 1.984.So the value of observed t is greater than tabulated value. Therefore the null hypothesis is rejected.

The assumptions for the test of this hypothesis is that the variance that has been used for the purpose of the test statistic is pooled variance. Pooled variance of the sample can be used if the sample variance for each population is assumed to be equal. In this case, the two standard deviations are unequal. So instead of pooled variance, one can use the following statistic:

The value of t statistic is then 11.66796.The value of t statistic is greater than observed value of t is 1.984, so the given hypothesis is rejected.

4:

The above test suggests that the mean value for the period before the financial crisis and the period after the financial crisis are different. The amount of loan redeemed after the financial crisis has been increased.

The test for whether the mean value of some credits redeemed before and after the year of the financial crisis has been done by taking the mean or the average values. It may often happen that the mean values are affected by outliers. To cater this problem, a robust measure has to be used test the hypothesis. For this, one can use the median test to determine whether the median value of the two tests are equal are not (Pan et al. 2014).

In the median test, the hypothesis of the test is, H0:me1=me2 against H1:me1<me2, where median 1 is the median for the sample of years before 2009 crisis and me 2 is the median of the years after 2009 financial crisis. The statistic for the test is:

Z is said to follow a standard normal distribution. So the statistic is rejected at 5% level of significance if the calculated value of z is greater than 1.645.

The calculated value of z is 135.9881.So the given hypothesis is rejected.That means the median of the two distribution are not equal(Brys, Hubert and Struyf 2012).

The mean value has been calculated by taking only nine years before the financial crisis. But that was the period of economic instability. So there were great fluctuations in the figures of loans redemption with interest. The test will be better if the sample is taken from the year 1990.So a test for the mean value has been done by taking the sample from the year 1990 to 2009 and 2010 to 2015, and the arithmetic mean values were compared with the help of t-test.The test statistic for the test is:

Where  Is the mean value from the year 1990 to 2009.The value of the t statistic calculated from the test -0.26599. The value of t observed from the t table at 0.05 % level of significance for degrees of freedom is 301 is 1.667.As the value of t observed is less than t tabulated, the given hypothesis is accepted. So this sample also reveals that the amount of loans redeemed post-financial crisis year has been increased.

References:

Altman, D., Machin, D., Bryant, T. And Gardner, M. Eds., 2013. Statistics with confidence: confidence intervals and statistical guidelines. John Wiley & Sons.

Aron, A., Coups, E. And Aron, E.N., 2013. Statistics for The Behavioral and Social Sciences: Pearson New International Edition: A Brief Course. Pearson Higher Ed.

Bera, A.K., Galvao, A.F. and Wang, L., 2014. On testing the equality of mean and quantile effects. Journal of Econometric Methods, 3(1), pp.47-62.

Bickel, P.J. and Lehmann, E.L., 2012. Descriptive statistics for nonparametric models IV. Spread. In Selected Works of EL Lehmann (pp. 519-526). Springer US.

Boone, H.N. and Boone, D.A., 2012. Analyzing likert data. Journal of extension, 50(2), pp.1-5.

Brys, G., Hubert, M. And Struyf, A., 2012. A robust measure of skewness.Journal of Computational and Graphical Statistics.

Hoenig, J.M. and Heisey, D.M., 2012. The abuse of power. The American Statistician.

KoÅ‚acz, A. And Grzegorzewski, P., 2016. Measures of dispersion for multidimensional data. European Journal of Operational Research, 251(3), pp.930-937.

Kruschke, J.K., 2013. Bayesian estimation supersedes the t test. Journal of Experimental Psychology: General, 142(2), p.573.

Pan, Y., Caudill, S.P., Li, R. And Caldwell, K.L., 2014. Median and quantile tests under complex survey design using SAS and R. Computer methods and programs in biomedicine, 117(2), pp.292-297.

Tablebuilder.singstat.gov.sg. (2016). Homepage | singstat Table Builder. [online] Available at: https://www.tablebuilder.singstat.gov.sg/publicfacing/mainmenu.action [Accessed 19 Aug. 2016].

Wan, X., Wang, W., Liu, J. And Tong, T., 2014. Estimating the sample mean and standard deviation from the sample size, median, range and/or interquartile range. BMC medical research methodology, 14(1), p.135.

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