Statistical Case Analysis

Question: Describe the descriptive statistical,relationship between education poverty and hypothesis testing? Answer: Introduction The document presents some statistical analysis based on data (presented in Appendix). The basic assumption taken for the task is that there should be some amount of relationship between education, poverty and per capita income. The more educated a society would be there would be higher chances of poverty alleviation. Such a place would have higher per capita income, even though here might be other factors influencing the same. Descriptive Statistical Analysis The section would investigate the nature of data with the statistical measures and graphs showing different attributes. This would help reader to understand the variations or symmetry in the data sets. Rural Per Income The rural per capita income of the Indian States has been shown in this section with the help of histogram. The histograms are the bar charts which show the variation in the range of data. The frequency along the vertical axis shows that the number of observations which fall in a particular range, which is plotted along the horizontal axis. The range of data (like per capita income) can be controlled according to the differences between the higher and lower values and the number of points required. The following histogram plot is more skewed towards he lower ranges of data (per capita income). This means that there more states with lower per capita income while lesser number of states with per capita income above the mean. In the following graph we may see that there are 7 states each in the per capita ranges of Rs. 895 to Rs. 995 and Rs. 995 to Rs.1095. While there are only one states each in the per capita ranges of Rs. 1195 to Rs. 1295 and Rs. 1295 to Rs.1395. Measures of Central Tendency The data in the below table, shows the statistical details of the per capita income in the rural areas across the Indian states. The mean per capita income is Rs. 928.73. This simply means that among the states this is the average value of the per capita income. The next value is the median which is Rs. 903.5 which is he value exactly in the middle of all data. The Mode is the data which has the highest frequency, here it is Rs. 930. The standard deviation of the data, 152.53, shows that there is high grade of variation. The low value of kurtosis shows that the distribution is quite close to the normal distribution. While the high skewness of data(0.77) shows that the data is asymmetric as has been discussed above. Per Capita Income Rural Areas in Indian States Mean 928.7333 Standard Error 27.84889 Median 903.5 Mode 930 Standard Deviation 152.5347 Sample Variance 23266.82 Kurtosis 0.2438 Skewness 0.772504 Range 606 Minimum 695 Maximum 1301 Sum 27862 Count 30 Urban Per Capita Income Similar to the plot of rural area per capita income, we obtain the histogram for the urban per capita income. In the following chart we see an opposite trend of the per capita income for urban areas with respect to per capita income in the rural areas. The incomes are more skewed towards the higher income. This means more number of Indian states can be found with higher per capita incomes than those with lower per capita incomes. Another feature of the following plot is that the per capita income of the urban areas is somewhat higher than the rural area and the spread of the following graph is lesser than the previous one. This means there are lesser extremes in per capita across the Indian saes in urban areas than the rural areas. Measures of Central Tendency The vital statistics for per capita income has been presented in the below table. The mean of per capita income is Rs. 1059.97. This is significantly higher than the mean per capita income of the rural areas. The median value is Rs. 1062, which simply means that half of the states would have per capita income higher than the value while have of the states, would have per capita income lower than the given value. The mode for the data sets is Rs. 1134, which is again quite higher than the per capita income of the rural counterpart. The standard deviation of the data is high (119.27) but lower than the value of rural area for per capita income. The skewness value is very low which points towards the symmetric distribution of the data. Similarly the value for kurtosis is quite low. Per Capita Income Urban Areas in Indian States Mean 1059.967 Standard Error 21.77716 Median 1062 Mode 1134 Standard Deviation 119.2784 Sample Variance 14227.34 Kurtosis -0.39729 Skewness 0.202935 Range 460 Minimum 849 Maximum 1309 Sum 31799 Count 30 Population below Poverty Line The histogram for the percentage of population below poverty line is quite different from the above plots as the plot below is not continuous and is bimodal distribution. This means that there are some states where the poverty line is lower while that are few states where the poverty line is quite higher (in terms of percentages). There are as many as 8 states where the percentage of population below poverty line is ranges from 12.09% to 15.09%. There are 4 states where the percentage of population below poverty line is higher than 33.09%. Measures of Central Tendency The mean percentage value for the people above poverty line is 19.07%. The value of mean poverty line is quite high. The median value is about 15.67%, which represent the middle value for the percentages above and below. The standard deviation of the data is low (10.95). The value of kurtosis is very high, this means that the distribution cannot be represented as a normal distribution. Percentage of Population of states below Poverty line Mean 19.07067 Standard Error 2.000949 Median 15.67 Mode #N/A Standard Deviation 10.95965 Sample Variance 120.1139 Kurtosis -1.15449 Skewness 0.604081 Range 34.84 Minimum 5.09 Maximum 39.93 Sum 572.12 Count 30 Number of Indian Universities Similar to the plot of percentage of population, below chart shows the number of Universities across the states. The graph is not continuous and the less number of Universities are more prevalent across the states. Few states are there which have number of Universities in the range of 57 to 62, while there are around 8 states with 7 to 12 Universities. Measures of Central Tendency The descriptive statistics show that the values in the following table. The mean for the number of Indian Universities is around 20. This is quite low figure for a country with large population. The median value is 17.5 which is close o the mean value. The mode value is surprisingly very low. The standard deviation is quite large comparing with the mean value. Even though the kurtosis is very low the histogram plot is hardly a normal plot. Number of Indian Universities Mean 20.53333 Standard Error 3.174877 Median 17.5 Mode 3 Standard Deviation 17.38952 Sample Variance 302.3954 Kurtosis -0.325 Skewness 0.869304 Range 57 Minimum 2 Maximum 59 Sum 616 Count 30 Relationship between Education Poverty We would try to find out if there is any relationship between the education and poverty. We would start with the assumption that education helps in uplifting population above the poverty line. In the above graph we have plotted the data of Universities and the data for poverty together. Since there is an inverse relationship between education and poverty, we see that both the curves are inverted with respect to each other, (like mirror images). The states with higher number of Universities should have less poverty percentage for the population. Regression Analysis The below tables show the regression values as calculated with help of excel. Multiple R value is very low; this means that here is very little correlation between the two data. This is again shown by the R square value which is very low (0.11). This is quite understandable result as we can see from the above plot that there is quite inverse relationship between the data. We see that both the curves are inverted with respect to each other, (like mirror images). SUMMARY OUTPUT Regression Statistics Multiple R 0.331764 R Square 0.110068 Adjusted R Square 0.078284 Standard Error 10.52192 Observations 30 ANOVA df SS MS F Significance F Regression 1 383.3986 383.3986 3.463062 0.073285 Residual 28 3099.905 110.7109 Total 29 3483.303 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0% Intercept 23.36403 3.002185 7.782345 1.77E-08 17.21434 29.51373 17.21434 29.51373 X Variable 1 -0.20909 0.112359 -1.86093 0.073285 -0.43925 0.021065 -0.43925 0.021065 Hypothesis Testing The poverty percentage for the last year was 21.9% for the whole country. Let us check whether the poverty decreased with respect to the previous year. Let us take the statement for the hypothesis as follows: We would use the student test to confirm whether the hypothesis can be rejected. The test is generally used in case of a small size sample. t-Test The t value can be calculated using the following formula. We would now calculate the t-value as -0.03297. The value from the t-table of corresponding is 0.127 (with 0.9). Thus, we can reject null hypothesis. This shows that there is a decrease in the poverty level form the previous year. Conclusion The data (as can be found in the appendix) shows different variables like per capita income of the rural and urban areas, the poverty level and Number of Universities of Indian States. The data suggest the variation of urban area is quite less as compared with the rural areas. There lower limit of the per capita income of the urban area is quite less than the rural areas. The distribution for the per capita income of both the rural and urban areas is skewed but normal. There should be some measurement to calculate the inverse dependence of the variables. The poverty and education has very inverse relationships, which is only partly proved by the regression analysis. The hypothesis test proves that there has been a decline in the poverty levels as compared with the previous years. Bibliography https://maps.unomaha.edu/. Week 3: Linear regression. https://maps.unomaha.edu/maher/GEOL2300/week3/week3.html (accessed January 10, 2015). Kean University. Regression and Correlation. https://www.kean.edu/~fosborne/bstat/09rc.html (accessed Jan 15, 2015). Stockburger, David W. Introductory Statistics: Concepts, Models, and Applications: REGRESSION MODELS. https://www.psychstat.missouristate.edu/introbook/sbk16.htm (accessed Janurary 18, 2015). Huff, D. (1993). How to Lie with Statistics. W. W. Norton Company. James, G. (2013). An Introduction to Statistical Learning: with Applications in R (Springer Texts in Statistics). Springer. Appendix Indian States Per capita Income Percentage below poverty Line Number of Universities Rural Urban Andhra Pradesh 860 1009 9.2 47 Arunachal Pradesh 930 1060 34.67 3 Assam 828 1008 31.98 9 Bihar 778 923 33.74 20 Chhattisgarh 738 849 39.93 3 Delhi 1145 1134 9.91 17 Goa 1090 1134 5.09 25 Gujarat 932 1152 16.63 2 Haryana 1015 1169 11.16 37 Himachal Pradesh 913 1064 8.06 22 Jammu Kashmir 891 988 10.35 18 Jharkhand 748 974 36.96 11 Karnataka 902 1089 20.91 12 Kerala 1018 987 7.05 43 Madhya Pradesh 771 897 31.65 17 Maharashtra 967 1126 17.35 33 Manipur 1118 1170 36.89 44 Meghalaya 888 1154 11.87 3 Mizoram 1066 1155 20.4 10 Nagaland 1270 1302 18.88 3 Odisha (Orissa) 695 861 32.59 4 Punjab 1054 1055 8.26 19 Rajasthan 905 1002 14.71 4 Sikkim 930 1226 8.19 19 Tamil Nadu 880 937 11.28 45 Tripura 798 920 14.05 6 Uttar Pradesh 880 1082 11.26 59 Uttarakhand 768 1082 29.43 3 West Bengal 783 981 19.98 58 Puducherry 1301 1309 9.69 20